Neighbour-transitive codes in Johnson graphs Mark Ioppolo Centre - - PowerPoint PPT Presentation

Codes in Graphs Neighbour-transitive codes

Neighbour-transitive codes in Johnson graphs

Mark Ioppolo

Centre for Mathematics of Symmetry and Computation University of Western Australia

August 6, 2013 Joint work with John Bamberg, Alice Devillers and Cheryl Praeger

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Coding Theory

Message: String of k elements from an alphabet Q. Encoder: Injective function Qk → Qn, where k ≤ n. An encoded message is called a codeword and the set of codewords is called a code.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Decoding and Error Correction

The Hamming metric counts the number of entries in which a pair of n−tuples disagree eg : d(000, 001) = 1. Decode based on this metric - nearest codeword decoding. δ(C) = Minimum distance of C. e = δ−1

2

• errors can be corrected.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Codes in Hamming graphs

Vertex set = Z3

2.

Vertices u, v are adjacent iff d(u, v) = 1. Example: Binary repetition code with δ = 3. 101 001 011 111 110 000 100 010

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Codes in Hamming graphs

Vertex set = Z3

2.

Vertices u, v are adjacent iff d(u, v) = 1. Example: Binary repetition code with δ = 3. 101 001 011 111 110 000 100 010

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Codes in graphs

Definition Let Γ be a graph. A code C in Γ is a subset of the vertex set of Γ. d(v1, v2) =length of shortest path from v1 to v2. Nearest codeword decoding still works. The minimum distance δ(C) is inherited from the graph metric.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Symmetry of codes

Perfect codes. Uniformly packed codes. Completely regular codes. Completely transitive. Neighbour-transitive. Definition Let C be a code in Γ. A code automorphism is an automorphism of Γ which stabilises C setwise.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Neighbour-transitive codes

Vertices adjacent to codewords which are not codewords themselves are called code-neighbours. Let C1 denote the neighbour set of C. 101 001 011 111 110 000 100 010 Definition A code is called G−neighbour-transitive if G ≤ Aut(C) and G acts transitively on C and C1.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Johnson Graphs

Let Ω be a set of size n. The Johnson graph J(Ω, k): Vertex set - {k element subsets of Ω}. Edge set - {pairs of vertices that intersect in k − 1 elements}.

Figure : J(5, 2)

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

NT-codes in Johnson graphs

Goal: Determine the G−neighbour-transitive codes in J(Ω, k). Aut(J(Ω, k)) = Sym(Ω) : k = 1

2|Ω|

2.Sym(Ω) : k = 1

2|Ω|

The action of G on Ω can be: Intransitive, Transitive and Imprimitive, or Primitive.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

First cases

The first two possibilities were studied by Bob Liebler and Cheryl Praeger. Intransitive: Correspond to completely regular codes of ‘strength zero’ Codes classified by Myerowitz. Transitive and imprimitive: Two classes built from a system of imprimitivity. Generalisation of Martin’s ‘groupwise complete designs’.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Strong incidence transitivity in J(Ω, k)

Let γ be a codeword and γ = Ω \ γ. Definition A code C is called G−strongly-incidence-transitive if G is transitive

• n C and the stabiliser Gγ of any codeword γ ∈ C is transitive on

γ × γ. g γ γ Ω Theorem (Liebler and Prager) If δ(C) ≥ 3 then strong-incidence-transitivity and neighbour-transitivity are equivalent concepts.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

SIT implies 2−transitive

Suppose G acts primitively. Theorem (Liebler and Prager) If G is primitive on Ω and strongly-incidence-transitive on codewords then G acts 2−transitively on Ω. g g Ω Thanks to the CFSG, we have a complete list of 2−transitive actions!

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

NT-codes with 2−transitive automorphism groups

Liebler and Prager: Suzuki, Ree, Unitary groups, Little Projective. Characterised affine and projective. Nico Durante: Affine groups, Projective Groups. Neunh¨

• ffer and Praeger:

Sporadic almost simple 2−transitive groups with δ > 3. Bamberg, Devillers, Praeger, MI: Binary symplectic groups acting on a set of quadratic forms.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Symplectic type NT-codes in J(Ω, k)

Let V = Z2n

2 . For x, y ∈ V define a symplectic form

B(x, y) = x1y2 + x2y1 + · · · + xn−1yn + xnyn−1. Let G = Isom(V , B) ∼ = Sp(2n, 2), and consider the set Q of quadratic forms which polarise to B: φ(x + y) + φ(x) + φ(y) = B(x, y).

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

The Jordan-Steiner Action

The group G = Sp(2n, 2) acts on Q by φg(x) := φ(xg). Theorem (Jordan, Steiner) The action of G on Q has two orbits, namely: Q+ : Hyperbolic quadratic forms. Q− : Elliptic quadratic forms. Moreover, G acts 2−transitively on each orbit. We consider codes C admitting automorphism group Aut(C) = Sp(2n, 2) in the Johnson graphs with Ω = Q± .

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Searching for Gγ

We would like a subgroup Gγ of G = Sp(2n, 2) which acts transitively on γ × γ. Gγ necessarily has 2 orbits on Ω. Lemma Let M ≤ G be a maximal subgroup containing Gγ. Then either M is intransitive and Gγ = M, or M is transitive and G = MGω.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Maximal Subgroups of Sp(2n, 2)

Aschbacher’s Theorem characterises the maximal subgroups of the classical groups: 8 geometric classes, or Modulo scalars is almost simple. Use the subgroup structure of the finite classical groups - Kleidman and Liebeck.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Aschbacher Class 1: Reducible Subgroups

M is the stabiliser of a subspace U ≤ V . M is the stabiliser of a chain 1 ≤ U ∩ U⊥ ≤ U ≤ U⊥ ≤ V where U⊥ is the set of vectors orthogonal to U. In this case, U must be either a non-degenerate U ∩ U⊥ = {0},

• r totally isotropic

U ⊆ U⊥.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Constructing a codeword: Totally isotropic case

Let V = Z2n

2 and B be an alternating bilinear form on V .

Choose a subspace U ≤ V such that U ⊆ U⊥. Theorem Define a codeword γ by γ = {φ ∈ Q±|U ⊆ Sing(φ)}. Then Gγ = GU and C = γG is a G−neighbour-transitive code in J(Q±, |γ|).

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Minimum Distance Computation

Let d = dim(U).

1 If ǫ = + and d < n then δ(C) = 22(n−1)−d 2 If ǫ = − and d < n − 1 then δ(C) = 22(n−1)−d − 2n−2 3 If ǫ = + and d = n or ǫ = − and d = n − 1 then δ(C) = 2d−1.

In particular, if n > 3 then all strongly-incidence-transitive codes are neighbour transitive. Moreover, neighbour-transitivity of codes with n ≤ 3 can be verified computationally.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

Constructing a codeword: Non-degenerate case

Let V = Z2n

2 and B be an alternating bilinear form on V .

Choose a non-degenerate subspace U ≤ V . Theorem (Liebler and Praeger) There exists an ǫ′ ∈ {+, −} such that γ consists of quadratic forms φ ∈ Qǫ and the restrictions φU, φU⊥ are of type ǫ′, ǫǫ′ respectively.

Mark Ioppolo Neighbour-transitive codes in Johnson graphs

Codes in Graphs Neighbour-transitive codes

37th Australasian Conference on Combinatorial Mathematics and Combinatorial Computing

December 9 – 13, 2013. UWA, Perth, Australia. 37accmcc.wordpress.com

Mark Ioppolo Neighbour-transitive codes in Johnson graphs