Special LCD Codes from Peisert and Generalized Peisert Graphs

Abstract

We examine binary and ternary codes from adjacency matrices of the Peisert graphs, $${\mathcal {P}}^*(q)$$ , and the generalized Peisert graphs, $$G{\mathcal {P}}^*(q)$$ , in particular those instances where the code is LCD and the dual of the code from the graph is the code from the reflexive graph. This occurs for all the binary codes and for those ternary codes for which $$q {\;\equiv 1}{ (\mathrm{mod}~3)}$$ . We find words of small weight in the codes, which, in the reflexive case, are likely to be minimum words. In addition we propose a decoding algorithm that can be feasible for these LCD codes.