Composite matrices from group rings, composite G-codes and constructions of self-dual codes

In this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters $$\gamma =7,8$$ γ = 7 , 8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other construction

Type IV codes over a non-unital ring

There is a special local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by [Formula: see text] We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over [Formula: see text] and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

New type i binary [72, 36, 12] self-dual codes from composite matrices and R1 lifts

<p style='text-indent:20px;'>In this work, we define three composite matrices derived from group rings. We employ these composite matrices to create generator matrices of the form <inline-formula><tex-math id="M3">\begin{document}$ [I_n \ | \ \Omega(v)], $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M4">\begin{document}$ I_n $\end{document}</tex-math></inline-formula> is the identity matrix and <inline-formula><tex-math id="M5">\begin{document}$ \Omega(v) $\end{document}</tex-math></inline-formula> is a composite matrix and search for binary self-dual codes with parameters <inline-formula><tex-math id="M6">\begin{document}$ [36,18, 6 \ \text{or} \ 8]. $\end{document}</tex-math></inline-formula> We next lift these codes over the ring <inline-formula><tex-math id="M7">\begin{document}$ R_1 = \mathbb{F}_2+u\mathbb{F}_2 $\end{document}</tex-math></inline-formula> to obtain codes whose binary images are self-dual codes with parameters <inline-formula><tex-math id="M8">\begin{document}$ [72,36,12]. $\end{document}</tex-math></inline-formula> Many of these codes turn out to have weight enumerators with parameters that were not known in the literature before. In particular, we find <inline-formula><tex-math id="M9">\begin{document}$ 30 $\end{document}</tex-math></inline-formula> new Type I binary self-dual codes with parameters <inline-formula><tex-math id="M10">\begin{document}$ [72,36,12]. $\end{document}</tex-math></inline-formula></p>