parallelisms of pg(3,4) with automorphisms of order 3acct2016/p46.pdf · anonymous (2, q +...

Parallelisms of PG(3,4) with

automorphisms of order 3

Svetlana Topalova, Stela Zhelezova

Institute of Mathematics and Informatics, BAS, Bulgaria

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Parallelisms of PG(3,4) with automorphisms of order 3

Parallelisms – relations and applications

Definitions and notations

History

PG(3,4) and related 2-designs

Construction

Results

Johnson, Combinatorics of Spreads and Parallelisms, CRC Press (2010)

constant dimension error correcting codes that contain lifted MRD codes: Etzion,

Silberstein, Codes and designs related to lifted MRD codes, 2011;

anonymous (2, q + 1)-threshold schemes: Stinson, Combinatorial designs:

constructions and analysis, 2004;

wireless key predistribution schemes: Ruj, Seberry, Roy, Key predistribution

schemes using block designs in wireless sensor networks, 2009.

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Parallelisms – relations and applications

Lunardon, On regular parallelisms in PG(3,q), 1984;

Walker, Spreads covered by derivable partial spreads, 1985;

Jha and Johnson, On regular r-packings, 1986 :

PG (2r − 1, q) and a translation plane of order q2r.

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Regular parallelisms – relations application

regular parallelism in

PG (3,q)

spread in PG(7,q)

translation plane of order q4

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Spread in PG(d,q) - a partition of the point set by lines.

Parallelism in PG(d,q) – a partition of the set of lines by spreads.

Automorphism group of the parallelism – maps each spread of the

parallelism to a spread of the same parallelism.

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Regulus of PG(3,q) – a set R={l1, …, lq+1} of mutually skew lines, any

line l li = pi,

l lj = pj, l ls ≠ Ø , ls R

l lk = pk,

Regular spread S={S1, …, Sq2+1} of PG(3,q) :

R(li, lj, lk) S.

Regular parallelism – all its spreads are regular.


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2-design:

V – finite set of v points

B – finite collection of b blocks: k-element subsets of V

D = (V, B ) – 2-(v,k,λ) design if any 2-subset of V is in λ blocks of B.

Parallel class – a partition of the point set by blocks.

Resolution – a partition of the collection of blocks by parallel classes.



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General constructions of parallelisms:

in PG(n,2), Zaicev, Zinoviev, Semakov, 1973; Baker, 1976.

in PG(2n-1,q), Beutelspacher, 1974.

a pair of orthogonal parallelisms – Fuji-Hara in PG(3,q), 1986.

two infinite families of regular cyclic parallelisms, PG(3,q), q ≡ 2

(mod 3), Penttila and Williams, 1998.

History

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Parallelisms in PG(3,q):

PG(3,2) – all (2) are classified, regular;

PG(3,3) – all (73 343) are classified, Betten, 2016;

PG(3,4) – with autom. of orders 5 or 7, Topalova, Zhelezova, 2013/2015;

PG(3,5) – cyclic parallelisms, Prince, 1998;

– regular parallelisms with autom. of order 3, Top., Zhel., 2016.

History


Regular parallelisms in PG(3,q):

PG(3,q) – two infinite families of regular cyclic parallelisms,

q ≡ 2 (mod 3), Penttila and Williams, 1998.

PG(3,2) – all are regular.

PG(3,5) – cyclic regular parallelisms, Prince, 1998

– regular parallelisms with autom. of order 3, Topalova, Zhelezova, 2016

(not from the Penttila and Williams families)

PG(3,3) – at most 11regular S in a parallelism, Betten, 2016.

PG(3,4) – at most 11regular S in a parallelism (order 5),

Topalova, Zhelezova, 2015.

History

Bamberg, 2012;

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The incidence of the points and t-dimensional subspaces of PG(d,q)

defines a 2-design (D).

points of D

blocks of D

resolutions of D

points of PG(3,4)

lines of PG(3,4)

parallelisms of PG(3,4)

2-(85,5,1) design


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PG(3,4) points, lines.

G – group of automorphisms of PG(3,4):

|G| = 213.34.52.7.17

Gi – subgroup of order i.


851

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q

qv

d

35711 22 qqq


t-dimentional subspaces 1 ( lines ) 2 (hyperplanes)

2-(v,k,) design 2-(85,5,1)

b=357, r=21

2-(85,21,5)

b=85, r=21

Parallelisms of PG(3,4)

21 spreads with 17 lines

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Sylow subgroup of order 81 (G81) - 3 conjugacy classes of its subgroups of

order 3 – GAP – http://www.gap-system.org

G3 – the subgroup of order 3 which yields parallelisms.

Construction

357 lines

27 fixed lines 110 orbits of

length 3

50 orbits with

nondisjoint lines

60 orbits with

disjoint lines

http://www.gap-system.org/









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Construction of spreads:

• backtrack search on lines;

• n+1st line - contains the first point, which is in none of the n spread lines;

• fixed spread – add the whole line orbit;

• non fixed spread – lines are from different orbits of one and the same length;

• lexicographically ordered;

• orbit leader – a fixed spread or the first in lexicographic order spread from an

orbit under G3;

Construction

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The types of orbit leaders under G3 :

• a spread of 2 fixed lines and 5 line orbits with disjoint lines (F2) – only one;

• a spread of 5 fixed lines and 4 line orbits with disjoint lines (F5) – 141 for each

fixed line containing the first point;

• a spread with an orbit of length three (O) – 29 624 disjoint to the fixed lines and

the F2 spread.

A parallelism invariant under G3

Construction

F2 F5 F5 F5 F5 F5 O O O O O

fixed part – 4959 nonisomorphic fixed parts

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Isomorphic solutions rejection

N (G3) – normalizer of G3 in G

| N (G3)| = 43 200

The rejection place

}|{)( 31

33 GggGGgGN

Construction

fixed part – 4959 nonisomorphic fixed parts

F2 F5 F5 F5 F5 F5 O O O O O

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R e s u l t s

|GP| 3 6 12 15 24 30 48 60 96 960

All 8 115 559 4 488 52 40 14 38 12 8 2 4

The order of the full automorphism group

of the parallelisms of PG(3,4) with automorphisms of order 3

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R e s u l t s

R S A 3 6 12 15 24 30 48 60 96 960 All

1* 0 5*15 259 661 434 16 12 260 123

1* 5* 15 13 052 250 20 10 13 332

1* 15 5* 3 886 116 12 4 014

1* 5*15 0 866 124 52 4 6 4 12 8 2 4 1 082

1*6 1*6 4*3 286 4 290

1*12 1*3 4* 10 10

other types 7 837 798 3 560 8 7 841 366

All 8 115 559 4 488 52 40 14 38 12 8 2 4 8 120 217

The type of the spreads

of the parallelisms of PG(3,4) with automorphisms of order 3

parallelisms of pg(3,4) with automorphisms of order 3acct2016/p46.pdf · anonymous (2, q +...

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