Self-orthogonal binary codes from odd graphs

Abstract

For k ≥ 2 and Ak an adjacency matrix for the odd graph Ok (of valency k + 1), we examine the row span C of Ak + I over F2 and find its dimension and minimum weight. We show that the self- orthogonal binary hull, C ∩ C⊥, has dimension (2k-1 k-1) - 2k-1 and is the binary code from an adjacency matrix M∗ of a graph OJk of valency (k+1)2 if k is odd, or from Mk+I if k is even. The symmetric group S2k+2 in its primitive representation of degreeacts as an automorphism group on these codes and on OJk. The graph OJk also arises from the orbit structure of the stabilizer of a point of S2k+2 in this representation. For k = 4 the dual of the binary hull is the code of a 2-(126,6,9) design on which S10 acts primitively on points, transitively on blocks. © 2015 Utilitas Mathematics.