Classes of self-orthogonal or self-dual codes from orbit matrices of Menon designs

Abstract

For every prime power q, where q≡1(mod4), and p a prime dividing q+12, we construct a self-orthogonal [2q,q−1] code and a self-dual [2q+2,q+1] code over the field of order p. The construction involves Paley graphs and the constructed [2q,q−1] and [2q+2,q+1] codes admit an automorphism group Σ(q) of the Paley graph of order q. If q is a prime and q=12m+5, where m is a non-negative integer, then the self-dual [2q+2,q+1]3 code is equivalent to a Pless symmetry code. In that sense we can view this class of codes as a generalization of Pless symmetry codes. For q=9 and p=5 we get a self-dual [20,10,8]5 code whose words of minimum weight form a 3-(20, 8, 28) design.