Optimal Three-Dimensional Optical Orthogonal Codes and Related Combinatorial Designs

Kenneth Shum

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Outline

• Three-dimensional optical orthogonal codes– Existing constructions– Bound on number of codewords

• Related combinatorial designs– Group divisible designs– Generalized Bhaskar Rao designs.

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1-D optical-orthogonal code• Spreading in time domain.• For two binary sequences x(t) and y(t) of length T, the

Hamming correlation function counts the number of overlapping ones after we cyclically shift y(t) by ,

• A (T, , a, c) 1-dimensional OOC is a set C of binary sequences of length T satisfying– Hxx(0) = , for all x in C, (constant weight)– Hxx() a, for all x in C and for all nonzero ,– Hxy() c, for distinct x and y in C, and all .

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2-D optical-orthogonal code• Spreading in time and frequency domains.• For two binary 2-dimensional arrays X(w,t) and Y(w,t) of size

WT, the Hamming correlation function counts the number of overlapping ones after we cyclically shift Y(w,t) in the second dimension by ,

• A (WT, , a, c) 2-dimensional OOC is a set C of binary arrays of size WT satisfying– HXX(0) = for all X in C, (constant weight)– HXX() a, for all X in C and for nonzero ,– HXY() c, for distinct X and Y, and all .

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3-D optical-orthogonal code• Spreading in spatial, frequency and time domains.• For two binary 3-dimensional arrays X(s,w,t) and Y(s,w,t) of

size SWT, the Hamming correlation function counts the number of overlapping ones after we cyclically shift Y(s,w,t) in the last dimension by ,

• A (SWT, , a, c) 3-dimensional OOC is a set C of binary arrays of size SWT satisfying– HXX(0) = for all X in C, (constant weight)– HXX() a, for all X in C and for nonzero ,– HXY() c, for distinct X and Y, and all .

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A small example• ( 222, 2, 0, 1) 3-D OOC

– S=2 spatial channels– W=2 wavelength– T=2 time chips in a period.– Hamming weight = 2– a = 0, zero auto-correlation– c = 1, cross-correlation upper bounded by 1.

First spatial plane

Second spatial plane

Code

wor

d 1 Co

dew

ord

2 Code

wor

d 3 Co

dew

ord

4 Code

wor

d 5 Co

dew

ord

6 Code

wor

d 7 Co

dew

ord

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wavelength 0wavelength 1

wavelength 0wavelength 1

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Different classes of 3-D OOC• A spatial plane is a wavelength/time plane (for a fixed index

of spatial channel).• At-most-one-pulse-per-plane code (AMOPPC)

– At most one optical pulse in each spatial plane.• Single-pulse-per-plane code (SPPC)

– Exactly one optical pulse in each spatial plane.• Multiple-pulse-per-plane code (MPPC)

– More than one optical pulse in each spatial plane.• At-most-one-pulse-per-time code (AMOPTC)

– Transmit at most one optical pulse in each time slot.• Single-pulse-per-time code (SPTC)

– Transmit exactly one optical pulse in each time slot.

a = 0

a = 0 S=

S

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Existing constructionsRef

.S, W, T |C| Type

[1] All prime factors of T SW SW 1 T MPPC

[1] All prime factors of W and T S S 1 W2T SPPC

[2] S = T = p, W = p2-1, p prime p2-1 1 p(p2-1) SPTC

[2] S = W = p, T = p2-1 p2-1 1 p2 SPTC

[3] S = W = T = p, p prime, 1 r p-2 S r W+1T SPPC

[3] S =4, W=q, T 2, q is a prime power 4 S 2 W3T2 SPPC

[3] S = q+1, W = q, T = p, q is a prime power 4, p is a prime q

3 1 W2T SPPC

[4] S = 3, W is even when T is even 3 1 W2T SPPC

[4] (S-1)WT even, 3|S(S-1)WTS0,1 mod 4 when T2 mod 4 and W1 mod

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3 1 AMOPPC

[1] Kim, Yu and Park, 2000.[2] Ortiz-Ubarri, Moreno and Tirkel, 2011.[3] Li, Fan and S, 2012.[4] S, 2013.

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Johnson-type bound

• For 3-D OOC in general, [2]

• For the class of at-most-one-pulse-per-plane code, [4]

[2] Ortiz-Ubarri, Moreno and Tirkel, 2011.[4] S, 2013.

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Perfect 3-D AMOPPC

• A 3-D at-most-one-pulse-per-plane code satisfying the second bound on code size is called perfect.

Remove all the floor operators

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Group divisible designs• Let v be a positive integer, K be a set of positive

integers, and be a positive integer.• A group divisible design GDD(K;v) of order v is and

block sizes from K is a triple (V,G,B) where– V is a set of size v, called points.– G is a partition of point set V, called groups,– B is a collection of subsets in V, called blocks, s.t.

• each block in B has size in K.• each block intersects every group in G in at most one point.• any pair of points from two distinct groups is contained in exactly

blocks of B.• The type of a GDD is the multi-set of group sizes.

– i.e., the multi-set {|H|: H G}.

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Example• v = 5, K = {2,3}, = 1.• V = {1,2,3,4,5}.• G = { {1}, {2,3}, {4,5} }• B = { {1,2,4}, {1,3,5}, {2,5}, {3,4} }

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2 3

4 5

Group 1

Group 2

Group 3Type = 1 , 22

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If all groups have size 1and all blocks have the same size, then GDDreduces to BIBD.

Generalized Bhaskar Rao design• Let G be a finite abelian group, and be a special symbol not in G.• A generalized Bhaskar Rao design, (n, k, ; G)-GBRD, is an

n b array with entries in G {}, such that– Each row has exactly r entries in G.– Each column contains exactly k entries in G.– Each pair of distinct rows (x1, x2, …, xb) and (y1, y2, …, yb), the list

xi – yi: i = 1,2,…,b, xi , yi ,contains exactly copies of each element of G.

• The parameters satisfy:– is a multiple of |G|.– bk = rn.– r(k –1) = (n – 1)

• If we replace by 0, andreplace group elements in G by 1,then what we get is the incident matrixof a balanced incomplete block design.

• The GBRD is said to be obtained bysigning the incidence matrix by G.

0 0 1 1 1 0 3 3 0

0 0 0 0 3 3 0 1 1

0 1 1 2 0 2 4 0 4

0 0 1 4 3 0 2 1 0

Example:(4, 3, 6; Z6)-GBRD

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0 1 1 1 0 1 1 1 0 1 1 1

1 0 1 1 1 0 1 1 1 0 1 1

1 1 0 1 1 1 0 1 1 1 0 1

1 1 1 0 1 1 1 0 1 1 1 0

Generalized Bhaskar Rao group divisible design

• If we start from a group divisible design, and sign the corresponding incidence matrix by the element in a finite group G, then the resulting matrix is called a generalized Bhaskar Rao group divisible design (GBRGDD)

• If we sign a GDD(K; ms) of type ms by abelian group G, the resulting GBRGDD is denoted by

(K, ; G)-GBRGDD of type ms.• if K is a singleton {k}, we write (k, ; G)-GBRGDD of

type ms.– Number of rows = ms.– Number of columns = s(s – 1)m2|G| / (k(k – 1)).

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Characterization of perfect 3-D OOC

Theorem: The followings are equivalent:• A perfect (SWT,,0,1)-AMOPPC.• (,T; ZT)-GBRGDD of type WS.

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First spatial plane

Second spatial plane

wavelength 0wavelength 1

wavelength 0wavelength 1

0 0 0 0

0 0 0 0

0 1 0 1

0 1 0 1

S=W=T==2 , =1

NEW

Existing result on existence of GDD

• Theorem [5]: Let m and s be positive integers. A necessary and sufficient condition for the existence of GDD(4; ms) of type ms is that the design is not GDD1(4; 8) of type 24, and not GDD1(4,24) of type 64, and such that

[5] Brouwer, Schrijver, and Hanani, 1977. (4, ;{e})-GBRGDD of type ms

(sm1, 4,0,1)-AMOPPC with T=1

Existing result on existence of GBRD

• For an integer g with prime factorization p1p2p3…pd, an elementary abelian group, EA(n), of order g is the direct product of Z/piZ, for i =1, 2 ,…, d.

• Theorem [6]: If g divides , then a (4,4,; EA(g))-GBRD exists unless– g 2 mod 4 when g is even,– g = = 3 when g is odd.

[6] Ge, Greig and Seberry, 2003. (4, ;EA(g))-GBRGDD of type 1s

(s1g, 4,0, )-AMOPPC with W=1

Product construction

Theorem [6]: Given a perfect (SW1T1, k, 0,1)-AMOPPC and a perfect (kW2T2, , 0,1)-AMOPPC, we can construct a perfect (S W1W2 T1T2, , 0,1)-AMOPPC.

[6] Ge, Greig and Seberry, 2003.

Proof by a product construction of GBRGDD in [6], translated to the setting ofAMOPPC by the one-to-one correspondence between GBRGDD over ZT AMOPPC.

New constructions of perfect 3-D OOCS, W, T |C| Type

S - 1 all prime power factors of WS all prime factors of T

S 1 W2T SPPC

(S-1)W 0 mod 3, S(S-1)W2 0 mod 12,

S 4, (4,2) (S,W) (4,6), and

T 5 be odd integer satisfyinggcd(T,27) 9, or

T = 9i for 5 i 31

4 1 SPPC

S = 4, T = 2, W even 4 1 W2T SPPC

S = 4, T even, T 4, W is even and WT 2e3 for any exponent e

4 1 W2T SPPC

T = 3, W not congruent to 2 mod 4 4 1 W2T SPPC

S = 4, T odd, T 5, 5 W 6,gcd(T,27) = 9, or T = 9i for 5 i 31

4 1 W2T SPPC

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