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 Q k → Q n , 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 . � δ − 1 � e = errors can be corrected. 2 Mark Ioppolo Neighbour-transitive codes in Johnson graphs
Codes in Graphs Neighbour-transitive codes Codes in Hamming graphs Vertex set = Z 3 2 . 011 111 Vertices u , v are adjacent iff 001 d ( u , v ) = 1. 101 010 Example: Binary repetition 110 code with δ = 3. 000 100 Mark Ioppolo Neighbour-transitive codes in Johnson graphs
Codes in Graphs Neighbour-transitive codes Codes in Hamming graphs Vertex set = Z 3 2 . 011 111 Vertices u , v are adjacent iff 001 d ( u , v ) = 1. 101 010 Example: Binary repetition 110 code with δ = 3. 000 100 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 ( v 1 , v 2 ) =length of shortest path from v 1 to v 2 . 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 C 1 denote the neighbour set of C . 011 111 001 101 010 110 100 000 Definition A code is called G − neighbour-transitive if G ≤ Aut ( C ) and G acts transitively on C and C 1 . 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 ). � Sym (Ω) : k � = 1 2 | Ω | Aut( J (Ω , k )) = 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 on C and the stabiliser G γ of any codeword γ ∈ C is transitive on γ × γ . Ω g γ γ 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 . Ω g g 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¨ offer 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 = Z 2 n 2 . For x , y ∈ V define a symplectic form B ( x , y ) = x 1 y 2 + x 2 y 1 + · · · + x n − 1 y n + x n y n − 1 . Let G = Isom( V , B ) ∼ = Sp(2 n , 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(2 n , 2) acts on Q by φ g ( x ) := φ ( x g ) . 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(2 n , 2) in the Johnson graphs with Ω = Q ± . Mark Ioppolo Neighbour-transitive codes in Johnson graphs
Codes in Graphs Neighbour-transitive codes Searching for G γ We would like a subgroup G γ of G = Sp(2 n , 2) which acts transitively on γ × γ . G γ necessarily has 2 orbits on Ω. Lemma Let M ≤ G be a maximal subgroup containing G γ . Then either M is intransitive and G γ = M, or M is transitive and G = MG ω . Mark Ioppolo Neighbour-transitive codes in Johnson graphs
Codes in Graphs Neighbour-transitive codes Maximal Subgroups of Sp(2 n , 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 } , or 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 = Z 2 n 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 G γ = G U and C = γ G 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 ) = 2 2( n − 1) − d 2 If ǫ = − and d < n − 1 then δ ( C ) = 2 2( n − 1) − d − 2 n − 2 3 If ǫ = + and d = n or ǫ = − and d = n − 1 then δ ( C ) = 2 d − 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 = Z 2 n 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 ∈ Q ǫ and the restrictions φ U , φ U ⊥ 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
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