Self-dual codes with an automorphism of order 13

Abstract

Using a method for constructing binary self-dual codes having an automorphism of odd prime order \begin{document}$p$\end{document} we classify, up to equivalence, all singly-even self-dual \begin{document}$[78,39,14]$\end{document} , \begin{document}$[80,40,14]$\end{document} , \begin{document}$[82,41,14],$\end{document} and \begin{document}$[84,42,14]$\end{document} codes as well as all doubly-even \begin{document}$[80,40,16]$\end{document} codes for \begin{document}$p=13$\end{document} . The results show that there are exactly 1592 inequivalent binary self-dual \begin{document}$[78,39,14]$\end{document} codes with an automorphism of type \begin{document}$13-(6,0)$\end{document} and we found 6 new values of the parameter in the weight function thus increasing more than twice the number of known values. As for binary \begin{document}$[80,40]$\end{document} self-dual codes with an automorphism of type \begin{document}$13-(6,2)$\end{document} there are 162696 singly-even self-dual codes with minimum distance 14 and 195 doubly-even such codes with minimum distance 16. We also construct many new codes of lengths 82 and 84 with minimum distance 14. Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before.