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

For every prime power q , where q ≡ 1 ( mod 4 ) , and p a prime dividing q + 1 2 , we construct a self-orthogonal [ 2 q , q − 1 ] code and a self-dual [ 2 q + 2 , q + 1 ] code over the field of order p . The construction involves Paley graphs and the constructed [ 2 q , q − 1 ] and [ 2 q + 2 , q + 1 ] codes admit an automorphism group Σ ( q ) of the Paley graph of order q . If q is a prime and q = 12 m + 5 , where m is a non-negative integer, then the self-dual [ 2 q + 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.