Permutation decoding of codes from generalized Paley graphs

The generalized Paley graphs GP(q,k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ GP }(q,k)$$\end{document} are a generalization of the well-known Paley graphs. Codes derived from the row span of adjacency and incidence matrices from Paley graphs have been studied in Ghinellie and Key (Adv Math Commun 5(1):93–108, 2011) and Key and Limbupasiriporn (Congr Numer 170:143–155, 2004). We examine the binary codes associated with the incidence designs of the generalized Paley graphs obtaining the code parameters [qs2,q-1,s]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\frac{qs}{2}, q-1, s]$$\end{document} or [qs,q-1,2s]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[qs, q-1,2s]$$\end{document} where s=q-1k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=\frac{q-1}{k}$$\end{document}. By finding explicit PD-sets we show that these codes can be used for permutation decoding.