Noncatastrophic convolutional codes over a finite ring

Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized as polynomial left prime matrices. In this paper, we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in the finite ring [Formula: see text]. In particular, we study the notion of zero left prime in order to fully characterize noncatastrophic encoders over the finite ring [Formula: see text]. The second part of the paper is devoted to investigate free and column distance of convolutional codes that are free finitely generated [Formula: see text]-modules. We introduce the notion of [Formula: see text]-degree and provide new bounds on the free distances and column distance. We show that this class of convolutional codes is optimal with respect to the column distance and to the free distance if and only if its projection on [Formula: see text] is.

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

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>

Linear codes over 𝔽4R and their MacWilliams identity

Let [Formula: see text] be the field of four elements. We denote by [Formula: see text] the commutative ring, with [Formula: see text] elements, [Formula: see text] with [Formula: see text]. This work defines linear codes over the ring of mixed alphabets [Formula: see text] as well as their dual codes under a nondegenerate inner product. We then derive the systematic form of the respective generator matrices of the codes and their dual codes. We wrap the paper up by proving the MacWilliams identity for linear codes over [Formula: see text].