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

İngilizce (OpenAlex)

We present a generator matrix of the form \begin{document}$ [ \sigma(v_1) \ | \ \sigma(v_2)] $\end{document} , where \begin{document}$ v_1 \in RG $\end{document} and \begin{document}$ v_2\in RH $\end{document} , for finite groups \begin{document}$ G $\end{document} and \begin{document}$ H $\end{document} of order \begin{document}$ n $\end{document} for constructing self-dual codes and linear complementary dual codes over the finite Frobenius ring \begin{document}$ R $\end{document} . In general, many of the constructions to produce self-dual codes forces the code to be an ideal in a group ring which implies that the code has a rich automorphism group. Unlike the traditional cases, codes constructed from the generator matrix presented here are not ideals in a group ring, which enables us to find self-dual and linear complementary dual codes that are not found using more traditional techniques. In addition to that, by using this construction, we improve \begin{document}$ 10 $\end{document} of the previously known lower bounds on the largest minimum weights of binary linear complementary dual codes for some lengths and dimensions. We also obtain \begin{document}$ 82 $\end{document} new binary linear complementary dual codes, \begin{document}$ 50 $\end{document} of which are either optimal or near optimal of lengths \begin{document}$ 41 \leq n \leq 61 $\end{document} which are new to the literature.