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Perfect Maps
Garth Isaak
Lehigh University
![Page 2: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/2.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
![Page 3: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/3.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
![Page 4: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/4.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
00
![Page 5: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/5.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
00, 01
![Page 6: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/6.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
00, 01, 11
![Page 7: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/7.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
00, 01, 11, 12
![Page 8: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/8.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
00, 01, 11, 12, 21
![Page 9: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/9.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
00, 01, 11, 12, 21, 10
![Page 10: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/10.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
00, 01, 11, 12, 21, 10, 02
![Page 11: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/11.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
00, 01, 11, 12, 21, 10, 02, 22
![Page 12: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/12.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
00, 01, 11, 12, 21, 10, 02, 22, 20
![Page 13: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/13.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
00, 01, 11, 12, 21, 10, 02, 22, 20
Each size 2 ternary string appears exactly once
Also called DeBruijn cycles
![Page 14: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/14.jpg)
A ternary 1-dimensional perfect map with window size 2
0 0 1 1 2 1 0 2 2 0 0 1 . . .
00, 01, 11, 12, 21, 10, 02, 22, 20
Each size 2 ternary string appears exactly once
Also called DeBruijn cycles
![Page 15: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/15.jpg)
History
? Other?
1892 E. Baudot - telegraphy- binary, window size 5
1894 C. Flye-Sainte Marie - monthly type questionGeneral construction and enumeration
1897 W. Mantel - primitive polynomials
1934 M.H. Martin - dynamics
1934 K.R. Popper - probability
1946 I.J. Good - decimal representation of numbers
1946 N.G. DeBruijn - telephone engineering
![Page 16: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/16.jpg)
History
? Other?
1892 E. Baudot - telegraphy- binary, window size 5
1894 C. Flye-Sainte Marie - monthly type questionGeneral construction and enumeration
1897 W. Mantel - primitive polynomials
1934 M.H. Martin - dynamics
1934 K.R. Popper - probability
1946 I.J. Good - decimal representation of numbers
1946 N.G. DeBruijn - telephone engineering
![Page 17: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/17.jpg)
History
? Other?
1892 E. Baudot - telegraphy- binary, window size 5
1894 C. Flye-Sainte Marie - monthly type questionGeneral construction and enumeration
1897 W. Mantel - primitive polynomials
1934 M.H. Martin - dynamics
1934 K.R. Popper - probability
1946 I.J. Good - decimal representation of numbers
1946 N.G. DeBruijn - telephone engineering
![Page 18: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/18.jpg)
History
? Other?
1892 E. Baudot - telegraphy- binary, window size 5
1894 C. Flye-Sainte Marie - monthly type questionGeneral construction and enumeration
1897 W. Mantel - primitive polynomials
1934 M.H. Martin - dynamics
1934 K.R. Popper - probability
1946 I.J. Good - decimal representation of numbers
1946 N.G. DeBruijn - telephone engineering
![Page 19: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/19.jpg)
History
? Other?
1892 E. Baudot - telegraphy- binary, window size 5
1894 C. Flye-Sainte Marie - monthly type questionGeneral construction and enumeration
1897 W. Mantel - primitive polynomials
1934 M.H. Martin - dynamics
1934 K.R. Popper - probability
1946 I.J. Good - decimal representation of numbers
1946 N.G. DeBruijn - telephone engineering
![Page 20: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/20.jpg)
History
? Other?
1892 E. Baudot - telegraphy- binary, window size 5
1894 C. Flye-Sainte Marie - monthly type questionGeneral construction and enumeration
1897 W. Mantel - primitive polynomials
1934 M.H. Martin - dynamics
1934 K.R. Popper - probability
1946 I.J. Good - decimal representation of numbers
1946 N.G. DeBruijn - telephone engineering
![Page 21: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/21.jpg)
History
? Other?
1892 E. Baudot - telegraphy- binary, window size 5
1894 C. Flye-Sainte Marie - monthly type question
General construction and enumeration
1897 W. Mantel - primitive polynomials
1934 M.H. Martin - dynamics
1934 K.R. Popper - probability
1946 I.J. Good - decimal representation of numbers
1946 N.G. DeBruijn - telephone engineering
![Page 22: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/22.jpg)
History
? Other?
1892 E. Baudot - telegraphy- binary, window size 5
1894 C. Flye-Sainte Marie - monthly type questionGeneral construction and enumeration
1897 W. Mantel - primitive polynomials
1934 M.H. Martin - dynamics
1934 K.R. Popper - probability
1946 I.J. Good - decimal representation of numbers
1946 N.G. DeBruijn - telephone engineering
![Page 23: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/23.jpg)
History
? Other?
1892 E. Baudot - telegraphy
- binary, window size 5
1894 C. Flye-Sainte Marie - monthly type questionGeneral construction and enumeration
1897 W. Mantel - primitive polynomials
1934 M.H. Martin - dynamics
1934 K.R. Popper - probability
1946 I.J. Good - decimal representation of numbers
1946 N.G. DeBruijn - telephone engineering
![Page 24: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/24.jpg)
History? Other?
1892 E. Baudot - telegraphy- binary, window size 5
1894 C. Flye-Sainte Marie - monthly type questionGeneral construction and enumeration
1897 W. Mantel - primitive polynomials
1934 M.H. Martin - dynamics
1934 K.R. Popper - probability
1946 I.J. Good - decimal representation of numbers
1946 N.G. DeBruijn - telephone engineering
![Page 25: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/25.jpg)
Notation
- I apologize, will try not to rely on this
notation too much
0 0 1 1 2 1 0 2 2 0 0 1 . . .
is in
PF31(9; 2; 1)
I PF - Perfect factor
I 3 - alphabet size 3
I 1 - dimension 1
I 9 - length 9
I 2 - window size 2
I 1 - 1 string
![Page 26: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/26.jpg)
Notation
- I apologize, will try not to rely on this
notation too much
0 0 1 1 2 1 0 2 2 0 0 1 . . .
is in
PF31(9; 2; 1)
I PF - Perfect factor
I 3 - alphabet size 3
I 1 - dimension 1
I 9 - length 9
I 2 - window size 2
I 1 - 1 string
![Page 27: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/27.jpg)
Notation
- I apologize, will try not to rely on this
notation too much
0 0 1 1 2 1 0 2 2 0 0 1 . . .
is in
PF31(9; 2; 1)
I PF - Perfect factor
I 3 - alphabet size 3
I 1 - dimension 1
I 9 - length 9
I 2 - window size 2
I 1 - 1 string
![Page 28: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/28.jpg)
Notation
- I apologize, will try not to rely on this
notation too much
0 0 1 1 2 1 0 2 2 0 0 1 . . .
is in
PF31(9; 2; 1)
I PF - Perfect factor
I 3 - alphabet size 3
I 1 - dimension 1
I 9 - length 9
I 2 - window size 2
I 1 - 1 string
![Page 29: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/29.jpg)
Notation - I apologize, will try not to rely on this
notation too much
0 0 1 1 2 1 0 2 2 0 0 1 . . .
is in
PF31(9; 2; 1)
I PF - Perfect factor
I 3 - alphabet size 3
I 1 - dimension 1
I 9 - length 9
I 2 - window size 2
I 1 - 1 string
![Page 30: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/30.jpg)
A1 = 0 0 0 1 2 2 1 2 1 0 0
A2 = 1 1 1 2 0 0 2 0 2 1 1
A3 = 2 2 2 0 1 1 0 1 0 2 2
is in
PF31(9; 3; 3)
![Page 31: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/31.jpg)
A1 = 0 0 0 1 2 2 1 2 1 0 0
A2 = 1 1 1 2 0 0 2 0 2 1 1
A3 = 2 2 2 0 1 1 0 1 0 2 2
is in
PF31(9; 3; 3)
Every ternary length 3 string appears exactly oncein this collection of 3 length 9 strings
![Page 32: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/32.jpg)
A1 = 0 0 0 1 2 2 1 2 1 0 0
A2 = 1 1 1 2 0 0 2 0 2 1 1
A3 = 2 2 2 0 1 1 0 1 0 2 2
is in
PF31(9; 3; 3)
Every ternary length 3 string appears exactly oncein this collection of 3 length 9 stringsFor example, 212 and 011 are indicated above
![Page 33: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/33.jpg)
0 0 0 1 00 0 1 0 01 0 1 1 10 1 1 1 00 0 0 1 0
is in PF22((4, 4); (2, 2); 1)
Every binary 2 by 2 array appears exactly once inthis 4 by 4, two dimensional array
![Page 34: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/34.jpg)
0 0 0 1 00 0 1 0 01 0 1 1 10 1 1 1 00 0 0 1 0
is in PF22((4, 4); (2, 2); 1)
Every binary 2 by 2 array appears exactly once in
this 4 by 4, two dimensional array For example1 00 1
and0 01 1
are indicated above (note the wrapping property)
![Page 35: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/35.jpg)
Review of Basics: A construction for 1-dimensional perfectmaps when k is a prime power: These are feedback shiftregister sequences
I Let h(x) = xn + hn−1xn−1 + · · ·+ h1x + x0 be a primitive
polynomial of degree n over GF (k)
I Let f (x1x2 . . . xn) = −h0x1 − h1x2 − · · · − hn−1xn
I Given terms in a string x1x2 . . . xn let the next term bef (x1x2 . . . xn)
I This produces a perfect map (except for omitting000 . . . 00)
I This method is useful for efficient construction and alsoused for 2-dimensional perfect factors ...(details omitted)
![Page 36: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/36.jpg)
Review of basics:
For all alphabet sizes k and window sizes n, one dimensionalperfect maps exist. That is, PFk
1(kn; n; 1) is non-empty. Notethat the string length is determined by k and n.Part of digraph D(3, 4):
2001
0010
0011
0012
0200
1200
2200
Construct a digraph D(k , n):Vertices ‘ = ‘ k-ary strings of length nArcs: (s1s2 . . . sn) −→ (s2s3 . . . snsn+1) between strings thatcan appear as consecutive windows
![Page 37: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/37.jpg)
Review of basics:For all alphabet sizes k and window sizes n, one dimensionalperfect maps exist. That is, PFk
1(kn; n; 1) is non-empty. Notethat the string length is determined by k and n.
Part of digraph D(3, 4):
2001
0010
0011
0012
0200
1200
2200
Construct a digraph D(k , n):Vertices ‘ = ‘ k-ary strings of length nArcs: (s1s2 . . . sn) −→ (s2s3 . . . snsn+1) between strings thatcan appear as consecutive windows
![Page 38: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/38.jpg)
Review of basics:For all alphabet sizes k and window sizes n, one dimensionalperfect maps exist. That is, PFk
1(kn; n; 1) is non-empty. Notethat the string length is determined by k and n.
Part of digraph D(3, 4):
2001
0010
0011
0012
0200
1200
2200
Construct a digraph D(k , n):Vertices ‘ = ‘ k-ary strings of length nArcs: (s1s2 . . . sn) −→ (s2s3 . . . snsn+1) between strings thatcan appear as consecutive windows
![Page 39: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/39.jpg)
Review of basics:For all alphabet sizes k and window sizes n, one dimensionalperfect maps exist. That is, PFk
1(kn; n; 1) is non-empty. Notethat the string length is determined by k and n.Part of digraph D(3, 4):
2001
0010
0011
0012
0200
1200
2200
Construct a digraph D(k , n):Vertices ‘ = ‘ k-ary strings of length nArcs: (s1s2 . . . sn) −→ (s2s3 . . . snsn+1) between strings thatcan appear as consecutive windows
![Page 40: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/40.jpg)
A Hamiltonian cycle in D(k, n)corresponds to a perfect map
000
100
010
001
111
110
101
011
![Page 41: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/41.jpg)
A Hamiltonian cycle in D(k, n)corresponds to a perfect map
000
100
010
001
111
110
101
011
000
![Page 42: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/42.jpg)
A Hamiltonian cycle in D(k, n)corresponds to a perfect map
000
001
100
010 111
110
101
011
0001
![Page 43: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/43.jpg)
A Hamiltonian cycle in D(k, n)corresponds to a perfect map
000
001 011
100
010 111
110
101
00011
![Page 44: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/44.jpg)
A Hamiltonian cycle in D(k, n)corresponds to a perfect map
000
001 011
111
100
010
110
101
000111
![Page 45: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/45.jpg)
A Hamiltonian cycle in D(k, n)corresponds to a perfect map
000
001 011
111
110100
010 101
0001110
![Page 46: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/46.jpg)
A Hamiltonian cycle in D(k, n)corresponds to a perfect map
000
001 011
111
110
101
100
010
00011101
![Page 47: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/47.jpg)
A Hamiltonian cycle in D(k, n)corresponds to a perfect map
000
001 011
111
110
101010
100
000111010
![Page 48: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/48.jpg)
A Hamiltonian cycle in D(k, n)corresponds to a perfect map
000
001 011
111
110
101010
100
0001110100
![Page 49: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/49.jpg)
A Hamiltonian cycle in D(k, n)corresponds to a perfect map
000
001 011
111
110
101010
100
00011101000
![Page 50: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/50.jpg)
Finding Hamiltonian cycles is ‘hard’
BUT ...
000
001 011
111
110
101010
100
is the line digraph of
00
10
01
11
100
001 011
110
000 111101 010
![Page 51: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/51.jpg)
Finding Hamiltonian cycles is ‘hard’BUT ...
000
001 011
111
110
101010
100
is the line digraph of
00
10
01
11
100
001 011
110
000 111101 010
![Page 52: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/52.jpg)
Finding Hamiltonian cycles is ‘hard’BUT ...
000
001 011
111
110
101010
100
is the line digraph of
00
10
01
11
100
001 011
110
000 111101 010
![Page 53: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/53.jpg)
Finding Eulerian circuits is ‘easy’
000
001 011
111
110
101010
100
00
10
01
11
100
001 011
110
000 111101 010
![Page 54: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/54.jpg)
Finding Eulerian circuits is ‘easy’
000
001 011
111
110
101010
100
000
00
10
01
11000
100
001 011
110
111101 010
![Page 55: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/55.jpg)
Finding Eulerian circuits is ‘easy’
000
001 011
111
110
101010
100
0001
00
10
01
11000001 01
1
111
110
101 010
100
![Page 56: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/56.jpg)
Finding Eulerian circuits is ‘easy’
000
001 011
111
110
101010
100
00011
00
10
01
11000001 01
1
111
110
101 010
100
![Page 57: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/57.jpg)
Finding Eulerian circuits is ‘easy’
000
001 011
111
110
101010
100
000111
00
10
01
11000001 01
1
111
110
101 010
100
![Page 58: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/58.jpg)
Finding Eulerian circuits is ‘easy’
000
001 011
111
110
101010
100
0001110
00
10
01
11000001 01
1
111
110
101 010
100
![Page 59: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/59.jpg)
Finding Eulerian circuits is ‘easy’
000
001 011
111
110
101010
100
00011101
00
10
01
11000001 01
1
111
110
101 010
100
![Page 60: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/60.jpg)
Finding Eulerian circuits is ‘easy’
000
001 011
111
110
101010
100
00011101 0
00
10
01
11000001 01
1
111
110
101 010
100
![Page 61: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/61.jpg)
Finding Eulerian circuits is ‘easy’
000
001 011
111
110
101010
100
00011101 00
00
10
01
11000001 01
1
111
110
101 010
100
![Page 62: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/62.jpg)
It is easy to check that the digraphs D(k , n − 1) are Eulerian:they are connected and each vertex has indegree andoutdegree k . The Eulerian circuits correspond to Hamiltoniancycles in D(k , n).
![Page 63: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/63.jpg)
Enumeration
BEST theorem: DeBruijn, Ehrenfest,Smith, Tutte:
Label the arcs leaving a given vertex in order that they aretraversed in the Eulerian circuit starting from 000 . . . 00. Thearcs traversed last form a spanning in-tree rooted at000 . . . 00. For each of the kn−1 vertices in D(k , n − 1) thereare (k − 1)! orderings for the arcs not in the tree. Thus the
number of Eulerian circuits is [(k − 1)!]kn−1
times the numberof trees rooted at some vertex. By the matrix tree theoremthe number of spanning trees is found by evaluating adeterminant of a matrix related to the incidence matrix andfor D(k − 1, n) this can be determined
![Page 64: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/64.jpg)
Enumeration BEST theorem: DeBruijn, Ehrenfest,Smith, Tutte:
Label the arcs leaving a given vertex in order that they aretraversed in the Eulerian circuit starting from 000 . . . 00. Thearcs traversed last form a spanning in-tree rooted at000 . . . 00. For each of the kn−1 vertices in D(k , n − 1) thereare (k − 1)! orderings for the arcs not in the tree. Thus the
number of Eulerian circuits is [(k − 1)!]kn−1
times the numberof trees rooted at some vertex. By the matrix tree theoremthe number of spanning trees is found by evaluating adeterminant of a matrix related to the incidence matrix andfor D(k − 1, n) this can be determined
![Page 65: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/65.jpg)
Enumeration BEST theorem: DeBruijn, Ehrenfest,Smith, Tutte:
Label the arcs leaving a given vertex in order that they aretraversed in the Eulerian circuit starting from 000 . . . 00. Thearcs traversed last form a spanning in-tree rooted at000 . . . 00.
For each of the kn−1 vertices in D(k , n − 1) thereare (k − 1)! orderings for the arcs not in the tree. Thus the
number of Eulerian circuits is [(k − 1)!]kn−1
times the numberof trees rooted at some vertex. By the matrix tree theoremthe number of spanning trees is found by evaluating adeterminant of a matrix related to the incidence matrix andfor D(k − 1, n) this can be determined
![Page 66: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/66.jpg)
Enumeration BEST theorem: DeBruijn, Ehrenfest,Smith, Tutte:
Label the arcs leaving a given vertex in order that they aretraversed in the Eulerian circuit starting from 000 . . . 00. Thearcs traversed last form a spanning in-tree rooted at000 . . . 00. For each of the kn−1 vertices in D(k , n − 1) thereare (k − 1)! orderings for the arcs not in the tree. Thus the
number of Eulerian circuits is [(k − 1)!]kn−1
times the numberof trees rooted at some vertex.
By the matrix tree theoremthe number of spanning trees is found by evaluating adeterminant of a matrix related to the incidence matrix andfor D(k − 1, n) this can be determined
![Page 67: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/67.jpg)
Enumeration BEST theorem: DeBruijn, Ehrenfest,Smith, Tutte:
Label the arcs leaving a given vertex in order that they aretraversed in the Eulerian circuit starting from 000 . . . 00. Thearcs traversed last form a spanning in-tree rooted at000 . . . 00. For each of the kn−1 vertices in D(k , n − 1) thereare (k − 1)! orderings for the arcs not in the tree. Thus the
number of Eulerian circuits is [(k − 1)!]kn−1
times the numberof trees rooted at some vertex. By the matrix tree theoremthe number of spanning trees is found by evaluating adeterminant of a matrix related to the incidence matrix andfor D(k − 1, n) this can be determined
![Page 68: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/68.jpg)
The number of perfect maps fork-ary windows of size n is
[(k − 1)!]kn−1
kkn−1−n
![Page 69: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/69.jpg)
Universal CyclesExtend perfect map ‘listing’ ideas to other combinatorialobjects.
Some recent non-existence results by Stevens forsubsets ...We illustrate with permutations (‘easier’ than subsets ...) :Look at length 3-permutations of {1, 2, 3, 4, 5}:
123421423215241321354153253452314531542543245134124351431251234
Every length 3-permutation of {1, 2, 3, 4, 5} appears exactlyonce
![Page 70: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/70.jpg)
Universal CyclesExtend perfect map ‘listing’ ideas to other combinatorialobjects. Some recent non-existence results by Stevens forsubsets ...
We illustrate with permutations (‘easier’ than subsets ...) :Look at length 3-permutations of {1, 2, 3, 4, 5}:
123421423215241321354153253452314531542543245134124351431251234
Every length 3-permutation of {1, 2, 3, 4, 5} appears exactlyonce
![Page 71: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/71.jpg)
Universal CyclesExtend perfect map ‘listing’ ideas to other combinatorialobjects. Some recent non-existence results by Stevens forsubsets ...We illustrate with permutations (‘easier’ than subsets ...) :Look at length 3-permutations of {1, 2, 3, 4, 5}:
123421423215241321354153253452314531542543245134124351431251234
Every length 3-permutation of {1, 2, 3, 4, 5} appears exactlyonce
![Page 72: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/72.jpg)
Universal CyclesExtend perfect map ‘listing’ ideas to other combinatorialobjects. Some recent non-existence results by Stevens forsubsets ...We illustrate with permutations (‘easier’ than subsets ...) :Look at length 3-permutations of {1, 2, 3, 4, 5}:
123421423215241321354153253452314531542543245134124351431251234
Every length 3-permutation of {1, 2, 3, 4, 5} appears exactlyonce
![Page 73: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/73.jpg)
Every length 3-permutation of {1, 2, 3, 4, 5} appears exactlyonce
1 2 3 4 2 1 4 2 3 5 2 1 . . .
123
![Page 74: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/74.jpg)
Every length 3-permutation of {1, 2, 3, 4, 5} appears exactlyonce
1 2 3 4 2 1 4 2 3 5 2 1 . . .
123, 234
![Page 75: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/75.jpg)
Every length 3-permutation of {1, 2, 3, 4, 5} appears exactlyonce
1 2 3 4 2 1 4 2 3 5 2 1 . . .
123, 234, 342
![Page 76: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/76.jpg)
Every length 3-permutation of {1, 2, 3, 4, 5} appears exactlyonce
1 2 3 4 2 1 4 2 3 5 2 1 . . .
123, 234, 342, 421
![Page 77: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/77.jpg)
Every length 3-permutation of {1, 2, 3, 4, 5} appears exactlyonce
1 2 3 4 2 1 4 2 3 5 2 1 . . .
123, 234, 342, 421, 214
![Page 78: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/78.jpg)
Every length 3-permutation of {1, 2, 3, 4, 5} appears exactlyonce
1 2 3 4 2 1 4 2 3 5 2 1 . . .
123, 234, 342, 421, 214, . . .
![Page 79: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/79.jpg)
Find a Hamiltonian cycle in a particular graph Q(n, k)
321
213
214
215
132
432
532
Graph for 3 permutations of {1, 2, 3, 4, 5}
Like the perfect map case these are line digraphs and similarmethods work to show existence of universal cycles for kpermutations of {1, 2, . . . , n}But Q(n, k) is the line digraph of some other digraph P(n, k)and not of Q(n, k − 1)
![Page 80: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/80.jpg)
Find a Hamiltonian cycle in a particular graph Q(n, k)
321
213
214
215
132
432
532
Graph for 3 permutations of {1, 2, 3, 4, 5}Like the perfect map case these are line digraphs and similarmethods work to show existence of universal cycles for kpermutations of {1, 2, . . . , n}
But Q(n, k) is the line digraph of some other digraph P(n, k)and not of Q(n, k − 1)
![Page 81: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/81.jpg)
Find a Hamiltonian cycle in a particular graph Q(n, k)
321
213
214
215
132
432
532
Graph for 3 permutations of {1, 2, 3, 4, 5}Like the perfect map case these are line digraphs and similarmethods work to show existence of universal cycles for kpermutations of {1, 2, . . . , n}But Q(n, k) is the line digraph of some other digraph P(n, k)and not of Q(n, k − 1)
![Page 82: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/82.jpg)
These other digraphs P(n, k) omit edges that do notcorrespond to permutations
321
213
214
215
132
432
532
Omit the red edges
Are these digraphs Hamiltonian?
![Page 83: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/83.jpg)
These other digraphs P(n, k) omit edges that do notcorrespond to permutations
321
213
214
215
132
432
532
Omit the red edgesAre these digraphs Hamiltonian?
![Page 84: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/84.jpg)
I Are these digraphs Hamiltonian?
I If yes then we get universal cycles fork-permutations for which the k + 1 strings arealso permutations
I These digraphs were introduced by Fiol et al. ina different context and the question ofHamiltonicity asked by Klerlein, Carr andStarling (at a Southeast conference)
![Page 85: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/85.jpg)
I Are these digraphs Hamiltonian?
I If yes then we get universal cycles fork-permutations for which the k + 1 strings arealso permutations
I These digraphs were introduced by Fiol et al. ina different context and the question ofHamiltonicity asked by Klerlein, Carr andStarling (at a Southeast conference)
![Page 86: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/86.jpg)
I Are these digraphs Hamiltonian?
I If yes then we get universal cycles fork-permutations for which the k + 1 strings arealso permutations
I These digraphs were introduced by Fiol et al. ina different context and the question ofHamiltonicity asked by Klerlein, Carr andStarling (at a Southeast conference)
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I These digraphs P(n, k) are NOT line digraphs
I BUT
I They are NEARLY line digraphs
I Obtain P(n, k) from the line digraph ofP(n, k − 1) by deleting a few arcs
I An Eulerian circuit in P(n, k) that avoids certainturns produces a Hamiltonian cycle in P(n, k)
![Page 88: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/88.jpg)
I These digraphs P(n, k) are NOT line digraphs
I BUT
I They are NEARLY line digraphs
I Obtain P(n, k) from the line digraph ofP(n, k − 1) by deleting a few arcs
I An Eulerian circuit in P(n, k) that avoids certainturns produces a Hamiltonian cycle in P(n, k)
![Page 89: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/89.jpg)
I These digraphs P(n, k) are NOT line digraphs
I BUT
I They are NEARLY line digraphs
I Obtain P(n, k) from the line digraph ofP(n, k − 1) by deleting a few arcs
I An Eulerian circuit in P(n, k) that avoids certainturns produces a Hamiltonian cycle in P(n, k)
![Page 90: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/90.jpg)
I These digraphs P(n, k) are NOT line digraphs
I BUT
I They are NEARLY line digraphs
I Obtain P(n, k) from the line digraph ofP(n, k − 1) by deleting a few arcs
I An Eulerian circuit in P(n, k) that avoids certainturns produces a Hamiltonian cycle in P(n, k)
![Page 91: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/91.jpg)
I These digraphs P(n, k) are NOT line digraphs
I BUT
I They are NEARLY line digraphs
I Obtain P(n, k) from the line digraph ofP(n, k − 1) by deleting a few arcs
I An Eulerian circuit in P(n, k) that avoids certainturns produces a Hamiltonian cycle in P(n, k)
![Page 92: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/92.jpg)
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Rearrange the ‘loops’ to avoid ‘bad turns’.
Construct a newdigraph on the loops with arcs for good turns. Find a Hamiltoniancycle to give the rearrangement. The Hamiltonian cycle exists inthe new digraph as degrees are high enough. Repeat for all verticesof P(n, k) to get a Hamiltonian cycle
![Page 101: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/101.jpg)
Rearrange the ‘loops’ to avoid ‘bad turns’. Construct a newdigraph on the loops with arcs for good turns.
Find a Hamiltoniancycle to give the rearrangement. The Hamiltonian cycle exists inthe new digraph as degrees are high enough. Repeat for all verticesof P(n, k) to get a Hamiltonian cycle
![Page 102: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/102.jpg)
Rearrange the ‘loops’ to avoid ‘bad turns’. Construct a newdigraph on the loops with arcs for good turns. Find a Hamiltoniancycle to give the rearrangement.
The Hamiltonian cycle exists inthe new digraph as degrees are high enough. Repeat for all verticesof P(n, k) to get a Hamiltonian cycle
![Page 103: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/103.jpg)
Rearrange the ‘loops’ to avoid ‘bad turns’. Construct a newdigraph on the loops with arcs for good turns. Find a Hamiltoniancycle to give the rearrangement. The Hamiltonian cycle exists inthe new digraph as degrees are high enough.
Repeat for all verticesof P(n, k) to get a Hamiltonian cycle
![Page 104: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/104.jpg)
Rearrange the ‘loops’ to avoid ‘bad turns’. Construct a newdigraph on the loops with arcs for good turns. Find a Hamiltoniancycle to give the rearrangement. The Hamiltonian cycle exists inthe new digraph as degrees are high enough. Repeat for all verticesof P(n, k) to get a Hamiltonian cycle
![Page 105: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/105.jpg)
I When n = k − 2 the degrees are not highenough to get a Hamiltonian cycle in theauxiliary digraph.
I In this case P(n, n − 2) are Cayley digraphsfrom an alternating group
I Use Rankin’s Theorem to conclude that there isno Hamiltonian cycle in this case
I Rankin’s Theorem 1948 application tocamponology (bell ringing)
![Page 106: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/106.jpg)
I When n = k − 2 the degrees are not highenough to get a Hamiltonian cycle in theauxiliary digraph.
I In this case P(n, n − 2) are Cayley digraphsfrom an alternating group
I Use Rankin’s Theorem to conclude that there isno Hamiltonian cycle in this case
I Rankin’s Theorem 1948 application tocamponology (bell ringing)
![Page 107: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/107.jpg)
I When n = k − 2 the degrees are not highenough to get a Hamiltonian cycle in theauxiliary digraph.
I In this case P(n, n − 2) are Cayley digraphsfrom an alternating group
I Use Rankin’s Theorem to conclude that there isno Hamiltonian cycle in this case
I Rankin’s Theorem 1948 application tocamponology (bell ringing)
![Page 108: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/108.jpg)
I When n = k − 2 the degrees are not highenough to get a Hamiltonian cycle in theauxiliary digraph.
I In this case P(n, n − 2) are Cayley digraphsfrom an alternating group
I Use Rankin’s Theorem to conclude that there isno Hamiltonian cycle in this case
I Rankin’s Theorem 1948 application tocamponology (bell ringing)
![Page 109: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/109.jpg)
Back to perfect maps
![Page 110: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/110.jpg)
Perfect Maps and Factors in higher dimensions - a partialhistory:
1961 Reed and Stewart
1985 Fan,Fan,Ma,Sui and Etzion
1988 Cock
1988 Ivanyi and Toth
1993 Hurlbert and Isaak
1994 Mitchell and Paterson
1996 Paterson
I and many others
![Page 111: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/111.jpg)
Necessary Conditions
R
S
u
v
I For 2-dimensional k−ary perfect maps
I
There are RS entries/windowsand kuv possible windows
I
So RS = kuv
I
The all 0 window is repeated if u = R or v = S
I
So R > u and S > v
![Page 112: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/112.jpg)
Necessary Conditions
R
S
u
v
I For 2-dimensional k−ary perfect maps
I There are RS entries/windowsand kuv possible windows
I
So RS = kuv
I
The all 0 window is repeated if u = R or v = S
I
So R > u and S > v
![Page 113: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/113.jpg)
Necessary Conditions
R
S
u
v
I For 2-dimensional k−ary perfect maps
I There are RS entries/windowsand kuv possible windows
I So RS = kuv
I
The all 0 window is repeated if u = R or v = S
I
So R > u and S > v
![Page 114: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/114.jpg)
Necessary Conditions
R
S
u
v
I For 2-dimensional k−ary perfect maps
I There are RS entries/windowsand kuv possible windows
I So RS = kuv
I The all 0 window is repeated if u = R or v = S
I
So R > u and S > v
![Page 115: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/115.jpg)
Necessary Conditions
R
S
u
v
I For 2-dimensional k−ary perfect maps
I There are RS entries/windowsand kuv possible windows
I So RS = kuv
I The all 0 window is repeated if u = R or v = S
I So R > u and S > v
![Page 116: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/116.jpg)
Necessary Conditions
RS = kuv and R > u, S > vSimilar conditions for perfect factors and for higher dimensions
I Are these sufficient?
I 1-dimensional perfect maps - YES
I 2-dimensional perfect maps when k is a prime power -YES (Paterson 1996)
I Otherwise? - partial results
I Difficulty with sizes like 212 × 312 with window size 3× 4and 6-ary
![Page 117: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/117.jpg)
Necessary Conditions
RS = kuv and R > u, S > vSimilar conditions for perfect factors and for higher dimensions
I Are these sufficient?
I 1-dimensional perfect maps - YES
I 2-dimensional perfect maps when k is a prime power -YES (Paterson 1996)
I Otherwise? - partial results
I Difficulty with sizes like 212 × 312 with window size 3× 4and 6-ary
![Page 118: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/118.jpg)
Necessary Conditions
RS = kuv and R > u, S > vSimilar conditions for perfect factors and for higher dimensions
I Are these sufficient?
I 1-dimensional perfect maps - YES
I 2-dimensional perfect maps when k is a prime power -YES (Paterson 1996)
I Otherwise? - partial results
I Difficulty with sizes like 212 × 312 with window size 3× 4and 6-ary
![Page 119: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/119.jpg)
Necessary Conditions
RS = kuv and R > u, S > vSimilar conditions for perfect factors and for higher dimensions
I Are these sufficient?
I 1-dimensional perfect maps - YES
I 2-dimensional perfect maps when k is a prime power -YES (Paterson 1996)
I Otherwise? - partial results
I Difficulty with sizes like 212 × 312 with window size 3× 4and 6-ary
![Page 120: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/120.jpg)
Necessary Conditions
RS = kuv and R > u, S > vSimilar conditions for perfect factors and for higher dimensions
I Are these sufficient?
I 1-dimensional perfect maps - YES
I 2-dimensional perfect maps when k is a prime power -YES (Paterson 1996)
I Otherwise? - partial results
I Difficulty with sizes like 212 × 312 with window size 3× 4and 6-ary
![Page 121: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/121.jpg)
Necessary Conditions
RS = kuv and R > u, S > vSimilar conditions for perfect factors and for higher dimensions
I Are these sufficient?
I 1-dimensional perfect maps - YES
I 2-dimensional perfect maps when k is a prime power -YES (Paterson 1996)
I Otherwise? - partial results
I Difficulty with sizes like 212 × 312 with window size 3× 4and 6-ary
![Page 122: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/122.jpg)
Non-prime powers alphabets from prime power alphabets:
0011 ∈ PF 12 (4; 2; 1)
{001, 112, 220} ∈ PF 13 (3; 2; 3)
‘Combine’
0 0 1 1 0 0 1 1 0 0 1 1⊕ 1 1 2 1 1 2 1 1 2 1 1 2
1 1 5 4 1 2 4 4 2 1 4 5
Using also 001 and 220 this gives 115412442145004301331034223520550253
∈ PF 16 (12; 2; 3)
![Page 123: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/123.jpg)
Higher Dimensional Perfect Maps
I Two basic techniques have been implicit inmany results
I ‘Integration’ ‘grows’ the window size
I ‘Concatenation’ increases the dimension
I Both use as tools perfect factors, perfectmultifactors, equivalence class perfectmultifactors ...
![Page 124: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/124.jpg)
Higher Dimensional Perfect Maps
I Two basic techniques have been implicit inmany results
I ‘Integration’ ‘grows’ the window size
I ‘Concatenation’ increases the dimension
I Both use as tools perfect factors, perfectmultifactors, equivalence class perfectmultifactors ...
![Page 125: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/125.jpg)
Higher Dimensional Perfect Maps
I Two basic techniques have been implicit inmany results
I ‘Integration’ ‘grows’ the window size
I ‘Concatenation’ increases the dimension
I Both use as tools perfect factors, perfectmultifactors, equivalence class perfectmultifactors ...
![Page 126: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/126.jpg)
Higher Dimensional Perfect Maps
I Two basic techniques have been implicit inmany results
I ‘Integration’ ‘grows’ the window size
I ‘Concatenation’ increases the dimension
I Both use as tools perfect factors, perfectmultifactors, equivalence class perfectmultifactors ...
![Page 127: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/127.jpg)
Higher Dimensional Perfect Maps
I Two basic techniques have been implicit inmany results
I ‘Integration’ ‘grows’ the window size
I ‘Concatenation’ increases the dimension
I Both use as tools perfect factors, perfectmultifactors, equivalence class perfectmultifactors ...
![Page 128: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/128.jpg)
Concatenation
Start with 1-dimensional perfect map 001121022 for columnsand shift sequence 01023456789 a perfect map with windowsize 1
0 1 2 3 4 5 6 7 8001121022
![Page 129: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/129.jpg)
Concatenation
Start with 1-dimensional perfect map 001121022 for columnsand shift sequence 01023456789 a perfect map with windowsize 1
0 1 2 3 4 5 6 7 80 00 01 11 12 21 10 02 22 2
![Page 130: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/130.jpg)
Concatenation
Start with 1-dimensional perfect map 001121022 for columnsand shift sequence 01023456789 a perfect map with windowsize 1
0 1 2 3 4 5 6 7 80 0 00 0 11 1 11 1 22 2 11 1 00 0 22 2 22 2 0
![Page 131: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/131.jpg)
Concatenation
Start with 1-dimensional perfect map 001121022 for columnsand shift sequence 01023456789 a perfect map with windowsize 1
0 1 2 3 4 5 6 7 80 0 0 10 0 1 21 1 1 11 1 2 02 2 1 21 1 0 20 0 2 02 2 2 02 2 0 1
![Page 132: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/132.jpg)
Concatenation
Start with 1-dimensional perfect map 001121022 for columnsand shift sequence 01023456789 a perfect map with windowsize 1
0 1 2 3 4 5 6 7 80 0 0 1 00 0 1 2 21 1 1 1 21 1 2 0 02 2 1 2 01 1 0 2 10 0 2 0 12 2 2 0 22 2 0 1 1
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Concatenation
Start with 1-dimensional perfect map 001121022 for columnsand shift sequence 01023456789 a perfect map with windowsize 1
0 1 2 3 4 5 6 7 80 0 0 1 0 00 0 1 2 2 11 1 1 1 2 11 1 2 0 0 22 2 1 2 0 11 1 0 2 1 00 0 2 0 1 22 2 2 0 2 22 2 0 1 1 0
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Concatenation
Start with 1-dimensional perfect map 001121022 for columnsand shift sequence 01023456789 a perfect map with windowsize 1
0 1 2 3 4 5 6 7 80 0 0 1 0 0 00 0 1 2 2 1 21 1 1 1 2 1 21 1 2 0 0 2 02 2 1 2 0 1 01 1 0 2 1 0 10 0 2 0 1 2 12 2 2 0 2 2 22 2 0 1 1 0 1
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Concatenation
Start with 1-dimensional perfect map 001121022 for columnsand shift sequence 01023456789 a perfect map with windowsize 1
0 1 2 3 4 5 6 7 80 0 0 1 0 0 0 10 0 1 2 2 1 2 21 1 1 1 2 1 2 11 1 2 0 0 2 0 02 2 1 2 0 1 0 21 1 0 2 1 0 1 20 0 2 0 1 2 1 02 2 2 0 2 2 2 02 2 0 1 1 0 1 1
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Concatenation
Start with 1-dimensional perfect map 001121022 for columnsand shift sequence 01023456789 a perfect map with windowsize 1
0 1 2 3 4 5 6 7 80 0 0 1 0 0 0 1 00 0 1 2 2 1 2 2 11 1 1 1 2 1 2 1 11 1 2 0 0 2 0 0 22 2 1 2 0 1 0 2 11 1 0 2 1 0 1 2 00 0 2 0 1 2 1 0 22 2 2 0 2 2 2 0 22 2 0 1 1 0 1 1 0
PF 23 ((9, 9); (2, 2); 1)
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Find1 20 2
where there is a shift of 2
The shift in location from 10 to 22 in 001121022
0 1 2 3 4 5 6 7 80 0 0 1 0 0 0 1 00 0 1 2 2 1 2 2 11 1 1 1 2 1 2 1 11 1 2 0 0 2 0 0 22 2 1 2 0 1 0 2 11 1 0 2 1 0 1 2 00 0 2 0 1 2 1 0 22 2 2 0 2 2 2 0 22 2 0 1 1 0 1 1 0
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I Concatenation increases the dimension
I In general needs 1-dimensional perfect factorsfor the shifts
I Concatenation of perfect factors requires two1-dimensional factors; one for shifts and one topick which factor
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I Concatenation increases the dimension
I In general needs 1-dimensional perfect factorsfor the shifts
I Concatenation of perfect factors requires two1-dimensional factors; one for shifts and one topick which factor
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I Concatenation increases the dimension
I In general needs 1-dimensional perfect factorsfor the shifts
I Concatenation of perfect factors requires two1-dimensional factors; one for shifts and one topick which factor
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‘Integrating’ to produce perfect factors (the inverseof Lempel’s homomorphism, finite differenceoperator):For 1-dimensional perfect factors:
0 0 1 1 2 1 0 2 20
The first row 001121022 is a PF 13 (9; 2; 1) (window size 2) and
gives the differences for (part of) a perfect factor with windowsize 3. Start with 0
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‘Integrating’ to produce perfect factors (the inverseof Lempel’s homomorphism, finite differenceoperator):For 1-dimensional perfect factors:
0 0 1 1 2 1 0 2 20 0
The first row 001121022 is a PF 13 (9; 2; 1) (window size 2) and
gives the differences for (part of) a perfect factor with windowsize 3
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‘Integrating’ to produce perfect factors (the inverseof Lempel’s homomorphism, finite differenceoperator):For 1-dimensional perfect factors:
0 0 1 1 2 1 0 2 20 0 0
The first row 001121022 is a PF 13 (9; 2; 1) (window size 2) and
gives the differences for (part of) a perfect factor with windowsize 3
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‘Integrating’ to produce perfect factors (the inverseof Lempel’s homomorphism, finite differenceoperator):For 1-dimensional perfect factors:
0 0 1 1 2 1 0 2 20 0 0 1
The first row 001121022 is a PF 13 (9; 2; 1) (window size 2) and
gives the differences for (part of) a perfect factor with windowsize 3
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‘Integrating’ to produce perfect factors (the inverseof Lempel’s homomorphism, finite differenceoperator):For 1-dimensional perfect factors:
0 0 1 1 2 1 0 2 20 0 0 1 2
The first row 001121022 is a PF 13 (9; 2; 1) (window size 2) and
gives the differences for (part of) a perfect factor with windowsize 3
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‘Integrating’ to produce perfect factors (the inverseof Lempel’s homomorphism, finite differenceoperator):For 1-dimensional perfect factors:
0 0 1 1 2 1 0 2 20 0 0 1 2 1
The first row 001121022 is a PF 13 (9; 2; 1) (window size 2) and
gives the differences for (part of) a perfect factor with windowsize 3
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‘Integrating’ to produce perfect factors (the inverseof Lempel’s homomorphism, finite differenceoperator):For 1-dimensional perfect factors:
0 0 1 1 2 1 0 2 20 0 0 1 2 1 2
The first row 001121022 is a PF 13 (9; 2; 1) (window size 2) and
gives the differences for (part of) a perfect factor with windowsize 3
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‘Integrating’ to produce perfect factors (the inverseof Lempel’s homomorphism, finite differenceoperator):For 1-dimensional perfect factors:
0 0 1 1 2 1 0 2 20 0 0 1 2 1 2 2
The first row 001121022 is a PF 13 (9; 2; 1) (window size 2) and
gives the differences for (part of) a perfect factor with windowsize 3
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‘Integrating’ to produce perfect factors (the inverseof Lempel’s homomorphism, finite differenceoperator):For 1-dimensional perfect factors:
0 0 1 1 2 1 0 2 20 0 0 1 2 1 2 2 1
The first row 001121022 is a PF 13 (9; 2; 1) (window size 2) and
gives the differences for (part of) a perfect factor with windowsize 3
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‘Integrating’ to produce perfect factors (the inverseof Lempel’s homomorphism, finite differenceoperator):For 1-dimensional perfect factors:
0 0 1 1 2 1 0 2 20 0 0 1 2 1 2 2 1
1 1 1 2 0 2 0 0 2
2 2 2 0 1 0 1 1 0
The top row 001121022 is a PF 13 (9; 2; 1) and gives the
differences for each of the other rows. The other rows differ bythe constant ‘starter’ in the first column.
The other rows form a PF 13 (9; 3; 3)
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0 0 1 1 2 1 0 2 20 0 0 1 2 1 2 2 11 1 1 2 0 2 0 0 22 2 2 0 1 0 1 1 0
Find 202: its differences are 12 so it will appear (in one of therows) in a location ‘below’ 12
Requires sum of the entries to be 0 mod k. If not then numberof factors will be decreased and length of each factor increasedWe will use integration along ‘directions’ for higherdimensional perfect factorsIn general for higher dimensions we need a perfect multifactorfor a ‘starter’
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0 0 1 1 2 1 0 2 20 0 0 1 2 1 2 2 11 1 1 2 0 2 0 0 22 2 2 0 1 0 1 1 0
Find 202: its differences are 12 so it will appear (in one of therows) in a location ‘below’ 12Requires sum of the entries to be 0 mod k. If not then numberof factors will be decreased and length of each factor increased
We will use integration along ‘directions’ for higherdimensional perfect factorsIn general for higher dimensions we need a perfect multifactorfor a ‘starter’
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0 0 1 1 2 1 0 2 20 0 0 1 2 1 2 2 11 1 1 2 0 2 0 0 22 2 2 0 1 0 1 1 0
Find 202: its differences are 12 so it will appear (in one of therows) in a location ‘below’ 12Requires sum of the entries to be 0 mod k. If not then numberof factors will be decreased and length of each factor increasedWe will use integration along ‘directions’ for higherdimensional perfect factors
In general for higher dimensions we need a perfect multifactorfor a ‘starter’
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0 0 1 1 2 1 0 2 20 0 0 1 2 1 2 2 11 1 1 2 0 2 0 0 22 2 2 0 1 0 1 1 0
Find 202: its differences are 12 so it will appear (in one of therows) in a location ‘below’ 12Requires sum of the entries to be 0 mod k. If not then numberof factors will be decreased and length of each factor increasedWe will use integration along ‘directions’ for higherdimensional perfect factorsIn general for higher dimensions we need a perfect multifactorfor a ‘starter’
![Page 155: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/155.jpg)
Perfect multifactors
000011210220112102201121022
is obtained by writing three 0’s followed by three copies of thestring 01121022. In this string every 3-ary window of length 2appears exactly 3 times, once in each position modulo 3. We callthis a perfect multifactor. Shifting by 3 and by 6 we get twoadditional strings
022000011210220112102201121 121022000011210220112102201
for a set of 3, length 27 strings in which each length 2 windowappears appears exactly once in each position modulo 9
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IntegrationUse the second string of the previous example and the 9× 9 arrayfrom the example preceding that:
0 2 2 0 0 0 0 1 1 2 1 0 2 2 0 1 1 2 1 0 2 2 0 1 1 2 10 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 00 0 1 2 2 1 2 2 1 0 0 1 2 2 1 2 2 1 0 0 1 2 2 1 2 2 11 1 1 1 2 1 2 1 1 1 1 1 1 2 1 2 1 1 1 1 1 1 2 1 2 1 11 1 2 0 0 2 0 0 2 1 1 2 0 0 2 0 0 2 1 1 2 0 0 2 0 0 22 2 1 2 0 1 0 2 1 2 2 1 2 0 1 0 2 1 2 2 1 2 0 1 0 2 11 1 0 2 1 0 1 2 0 1 1 0 2 1 0 1 2 0 1 1 0 2 1 0 1 2 00 0 2 0 1 2 1 0 2 0 0 2 0 1 2 1 0 2 0 0 2 0 1 2 1 0 22 2 2 0 2 2 2 0 2 2 2 2 0 2 2 2 0 2 2 2 2 0 2 2 2 0 22 2 0 1 1 0 1 1 0 2 2 0 1 1 0 1 1 0 2 2 0 1 1 0 1 1 0
Integrate down the columns:
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integration down the columns yields:
0 2 2 0 0 0 0 1 1 2 1 0 2 2 0 1 1 2 1 0 2 2 0 1 1 2 10 2 2 1 0 0 0 2 1 2 1 0 0 2 0 1 2 2 1 0 2 0 0 1 1 0 10 2 0 0 2 1 2 1 2 2 1 1 2 1 1 0 1 0 1 0 0 2 2 2 0 2 21 0 1 1 1 2 1 2 0 0 2 2 0 0 2 2 2 1 2 1 1 0 1 0 2 0 02 1 0 1 1 1 1 2 2 1 0 1 0 0 1 2 2 0 0 2 0 0 1 2 2 0 21 0 1 0 1 2 1 1 0 0 2 2 2 0 2 2 1 1 2 1 1 2 1 0 2 2 02 1 1 2 2 2 2 0 0 1 0 2 1 1 2 0 0 1 0 2 1 1 2 0 0 1 02 1 0 2 0 1 0 0 2 1 0 1 1 2 1 1 0 0 0 2 0 1 0 2 1 1 21 0 2 2 2 0 2 0 1 0 2 0 1 1 0 0 0 2 2 1 2 1 2 1 0 1 1
Doing the same thing with the other two possible starters producesthree 3-ary 9× 27 arrays in which we claim that every 3-ary 3× 2subarray appears exactly once.
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The examples we have given hint at several general methods whichgive hope that that necessary conditions can be shown sufficient inhigher dimensions at least for prime power alphabets.
For non prime power alphabet sizes new tools will probably beneeded
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The examples we have given hint at several general methods whichgive hope that that necessary conditions can be shown sufficient inhigher dimensions at least for prime power alphabets.For non prime power alphabet sizes new tools will probably beneeded
![Page 160: Perfect Maps - Lehighgi02/mcccf.pdf1934 M.H. Martin - dynamics 1934 K.R. Popper - probability 1946 I.J. Good - decimal representation of numbers 1946 N.G. DeBruijn - telephone engineering](https://reader036.vdocuments.site/reader036/viewer/2022071605/6141f2be2035ff3bc7625b3a/html5/thumbnails/160.jpg)
Let A be a (~R; ~V ; τ)dG [~N] PMF (perfect multifactor). LetH = Zr1/n1
× Zr2/n2× · · ·Zrd/nd
and let H ′ = {1, 2, . . . , τ}. Let(B : C ) be a(Q; (U − 1,U); ρ)H,H′ [M] PMFP (perfect multifactor pair) withthe following property. There exists c ∈ H such that each stringB(j) in B satisfiesQ∑
h=1
[B(j)]h = c . That is, the entries in each fundamental block sum
to c .Then,
I If c = 0 ∈ H, concatenation using (B : C ) as indexer yields a(~R+; ~V +; ρ)d+1
G [~N+] PMF (perfect multifactor) where the
first d coordinates of ~N+, ~R+ and ~V + are the same as ~N, ~Rand ~V and n+
d+1 = M, r+d+1 = Q and v+
d+1 = U.I If c 6= 0 ∈ H and additionally we have the following: If c is
viewed as a vector ~C = (c1, c2, . . . , cd) with entries from Z
and for i = 1, 2, . . . , d we have ηi =ri/ni
gcd(ri/ni , ci )(i.e., the
order of ci in Zri/niis ηi ), then gcd (ηi , ci ) = 1. Also, for i 6= j ,
gcd(ηi , ηj) = 1. Then concatenation using (B : C ) as indexer
yields a (~R+; ~V +; ρ)d+1G [~N∗] PMF (perfect multifactor) where
the first d coordinates of ~R+ and ~V + are the same as ~R and~V with r+
d+1 = QΠdi=1ηi and v+
d+1 = U. Also ~N∗ is given byn∗j = niηi for j = 1, 2, . . . , d and n∗
d+1 = M.
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Let A be a (~Q; ~U; ρ)dG [~N] PMF (perfect multifactor) with the sumof entries in each (one dimensional) projection along direction dequal to a constant c ∈ G . Let ~Q− and ~U− be obtained from ~Qand ~U by deleting the d th dimension.Then,
I If c = 0, let B be a (~R; ~U−; τ)d−1G |H [~Q−] EPMF (equivalence
class perfect multifactor modulo H). Integrating A withstarter B yields a(~R+; ~U∗; ρτ)dG |H [~N] EPMF (equivalence class perfect
multifactor modulo H) where ~U∗ = ~U + ~e(d) and r+d = qd .
I If c 6= 0, let H ′ be the subgroup generated by c . Let B be aset of representatives modulo H ′ of a (~R; ~U−; τ)d−1
G |H′ [~Q−]
EPMF (equivalence class perfect multifactor modulo H ′).Integrating A with starter B yields a (~R+; ~U∗; ρτ/|H ′|)dG [~N]
PMF (perfect multifactor) where ~U∗ = ~U + ~e(d) andr+d = |H ′|qd .