The construction of combinatorial structures and linear codes from orbit matrices of strongly regular graphs

Transcription

1 linear codes from Sanja Rukavina Department of Mathematics University of Rijeka, Croatia Symmetry vs Regularity The first 50 years since Weisfeiler-Leman stabilization July - July 7, 08, Pilsen, Czech Republic Supported by CSF under the project 637 /

2 OM of M Behbahani and C Lam have studied strongly regular that admit an automorphism group of prime order M Behbahani, C Lam, Strongly regular with non-trivial automorphisms, Discrete Math, 3 (0), 3- Let Γ be a srg(v, k, λ, µ) and A be its adjacency matrix Suppose an automorphism group G of Γ partitions the set of vertices V into t orbits O,, O t, with sizes n,, n t, respectively The orbits divide A into submatrices [A ij ], where A ij is the adjacency matrix of vertices in O i versus those in O j We define matrices C = [c ij ] and R = [r ij ], i, j t, such that c ij = column sum of A ij, r ij = row sum of A ij R is related to C by r ij n i = c ij n j Since the adjacency matrix is symmetric, R = C T The matrix R is the row orbit matrix of the graph Γ with respect to G, and the matrix C is the column orbit matrix of the graph Γ with respect to G /

3 srg(0,3,0,) R = C = /

4 Definition A (t t)-matrix R = [r ij ] with entries satisfying conditions t ns t t n i r ij = r ij = k () j= i= n j r si r sj = δ ij (k µ) + µn i + (λ µ)r ji () s= n j is called a row orbit matrix for a graph with parameters (v, k, λ, µ) and orbit lengths distribution (n,, nt ) A (t t)-matrix C = [c ij ] with entries satisfying conditions t ns t t n j c ij = c ij = k (3) i= j= n i c is c js = δ ij (k µ) + µn i + (λ µ)c ij () s= n j is called a column orbit matrix for a graph with parameters (v, k, λ, µ) and orbit lengths distribution (n,, nt ) If all orbits have the same length w, ie n i = w for i =,, t, then C = R, and the following holds t r is r js = δ ij (k µ) + µw + (λ µ)r ij s= /

5 A code C of length n over the alphabet Q is a subset C Q n Elements of a code are called codewords A code C is called a p-ary linear code of dimension m if Q = F p, for a prime p, and C is an m-dimensional subspace of a vector space F n p Let C F n p be a linear code Its dual code is the code C = {x F n p x c = 0, c C}, where is the standard inner product A code C is self-orthogonal if C C and self-dual if C = C Theorem [ D Crnković, M Maksimović, B G Rodrigues, SR, 06] Let Γ be a srg(v, k, λ, µ) with an automorphism group G which acts on the set of vertices of Γ with v w orbits of length w Let R be the row orbit matrix of the graph Γ with respect to G If q is a prime dividing k, λ and µ, then the matrix R generates a self-orthogonal code of length v w over F q 5 /

6 Theorem [ D Crnković, M Maksimović, SR, 08] Let Γ be a SRG(v, k, λ, µ) having an automorphism group G which acts on the set of vertices of Γ with b orbits of lengths n,, n b, respectively, with f fixed vertices, and the other b f orbits of lengths n f +,, n b divisible by p, where p is a prime dividing k, λ and µ Let C be the column orbit matrix of the graph Γ with respect to G If q is a prime power such that q = p n, then the code spanned by the rows of the fixed part of the matrix C is a self-orthogonal code of length f over F q C n f + n b n f + n b 6 /

7 Theorem [ D Crnković, M Maksimović, SR, 08] Let Γ be a SRG(v, k, λ, µ) having an automorphism group G which acts on the set of vertices of Γ with b orbits of lengths n,, n b, respectively, such that there are f fixed vertices, h orbits of length w, and b f h orbits of lengths n f +h+,, n b Further, let pw n s if w < n s, and pn s w if n s < w, for s = f + h +,, b, where p is a prime number dividing k, λ, µ and w Let C be the column orbit matrix of the graph Γ with respect to G If q is a prime power such that q = p n, then the code over F q spanned by the part of the matrix C (rows and columns) determined by the orbits of length w is a self-orthogonal code of length h C w w n f +h+ n b w w n f +h+ n b 7 /

8 C C C 8 /

9 Theorem [ D Crnković, M Maksimović, SR, 08] Let Γ be a SRG(v, k, λ, µ) with an automorphism group G which acts on the set of vertices of Γ with b orbits of lengths n,, n b, respectively, and w = max{n,, n b } Further, let p be a prime dividing k, λ, µ and w, and let pn s w if n s w Let C be the column orbit matrix of the graph Γ with respect to G If q is a prime power such that q = p n, then the code over F q spanned by the rows of C corresponding to the orbits of length w is a self-orthogonal code of length b C n n i n i+ n i w w n n i n i+ n i w w 9 /

10 Theorem [ D Crnković, M Maksimović, SR, 08] Let Γ be a SRG(v, k, λ, µ) with an automorphism group G which acts on the set of vertices of Γ with b orbits of lengths n,, n b, respectively, and w = min{n,, n b } Further, let p be a prime dividing k, λ, µ and w, and let pw n s if n s w Let R be the row orbit matrix of the graph Γ with respect to G If q is a prime power such that q = p n, then the code over F q spanned by the rows of R corresponding to the orbits of length w is a self-orthogonal code of length b R w w n i+ n i n il + n b w w n i+ n i n il + n b 0 /

11 Theorem [ D Crnković, SR, A Švob, 08] Let C = [c ij ] be a (t t) column orbit matrix for a graph Γ with parameters (v, k, λ, µ) and orbit lengths distribution (n,, n t ), n = = n t = n, with constant diagonal Further, let the off-diagonal entries of C have exactly two values, ie c ij {x, y}, x y, i, j t, i j Replacing every x with and all diagonal elements and every y in C with 0, one obtains the adjacency matrix of a graph Γ on t vertices /