Block designs from self-dual codes obtained from Paley designs and Paley graphs

In 2002, P. Gaborit introduced two constructions of self-dual codes using quadratic residues, so-called pure and bordered construction, as a generalization of the Pless symmetry codes. In this paper, we further study conditions under which the pure and the bordered construction using Paley designs and Paley graphs yield self-dual codes. Special attention is given to the binary and ternary codes. Further, we construct \(t\)-designs from supports of the codewords of a particular weight in the binary and ternary codes obtained.