Binary self-dual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme

Abstract

<p style='text-indent:20px;'>We present a generator matrix of the form <inline-formula><tex-math id="M1">\begin{document}$ [ \sigma(v_1) \ | \ \sigma(v_2)] $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ v_1 \in RG $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ v_2\in RH $\end{document}</tex-math></inline-formula>, for finite groups <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ H $\end{document}</tex-math></inline-formula> of order <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula> for constructing self-dual codes and linear complementary dual codes over the finite Frobenius ring <inline-formula><tex-math id="M7">\begin{document}$ R $\end{document}</tex-math></inline-formula>. In general, many of the constructions to produce self-dual codes forces the code to be an ideal in a group ring which implies that the code has a rich automorphism group. Unlike the traditional cases, codes constructed from the generator matrix presented here are not ideals in a group ring, which enables us to find self-dual and linear complementary dual codes that are not found using more traditional techniques. In addition to that, by using this construction, we improve <inline-formula><tex-math id="M8">\begin{document}$ 10 $\end{document}</tex-math></inline-formula> of the previously known lower bounds on the largest minimum weights of binary linear complementary dual codes for some lengths and dimensions. We also obtain <inline-formula><tex-math id="M9">\begin{document}$ 82 $\end{document}</tex-math></inline-formula> new binary linear complementary dual codes, <inline-formula><tex-math id="M10">\begin{document}$ 50 $\end{document}</tex-math></inline-formula> of which are either optimal or near optimal of lengths <inline-formula><tex-math id="M11">\begin{document}$ 41 \leq n \leq 61 $\end{document}</tex-math></inline-formula> which are new to the literature.</p>