Construction of self dual codes from graphs

In this work we define and study binary codes \(C_{q,k}\) and \(\overline{C_{q,k}}\) obtained from neighborhood designs of Paley-type bipartite graphs P(q, k) and their complements, respectively for q an odd prime. We prove that for some values of q and k the codes \({C}_{q,k}\) are self-dual and the codes \(\overline{C_{q,k}}\) are self-orthogonal. Most of these codes tend to be with optimal or near optimal parameters. Next, we extend the codes \(C_{q,k}\) to get doubly even self dual codes and find that most of these codes are extremal.