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

For every prime power $q$, where $q \equiv 1\ (mod\ 4)$, and $p$ a prime dividing $\frac{;q+1};{;2};$, 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 $\Sigma (q)$ 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.

self-orthogonal code ; self-dual code ; Pless symmetry code ; Paley graph ; block design

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