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

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