arXiv subjects

Mohammed El Oued

Publications and source records attributed to Mohammed El Oued.

3 recordsLinked to original sources

arXiv2026

An Algebraic Invariant for Free Convolutional Codes over Finite Local Rings

This paper investigates the algebraic structure of free convolutional codes over the finite local ring Z_{p^r}. We introduce a new structural invariant, the Residual Structural Polynomial, denoted by Delta_p(C) in F_p[D]. We construct this invariant via encoders which are reduced internal degree matrices (RIDM). We formally demonstrate that Delta_p(C) is an intrinsic characteristic of the code, invariant under equivalent RIDMs. A central result of this work is the establishment that Delta_p(C) serves as an algebraic criterion for intrinsic catastrophicity: we prove that a free code C admits a non-catastrophic realization if and only if Delta_p(C) is a monomial of the form D^s. Furthermore, we establish a fundamental duality theorem, proving that Delta_p(C) = Delta_p(C^perp). This result reveals a deep structural symmetry, showing that the "catastrophicity" of a free code is preserved under orthogonality.

cs.IT↗

arXiv2025

On catastrophicity of convolutional codes and their encoders over $\Z_{p^r}$

This paper investigates the existence of minimal $p$-encoders for convolutional codes over $\mathbb{Z}_{p^r}$, where $p$ is a prime. This addresses a conjecture from \cite{k}, which posits that every such code admits a minimal $p$-encoder, implying that all convolutional codes over $\mathbb{Z}_{p^r}$ are noncatastrophic when input sequences are restricted to coefficients in $\{0, \dots, p-1\}$. Our contributions include the introduction of a new polynomial invariant that characterizes free codes, which enables us to establish a necessary and sufficient condition for a free code over $\mathbb{Z}_{p^r}$ to be noncatastrophic in the usual sense (where input coefficients are from $\mathbb{Z}_{p^r}$). Based on these findings, we affirm the conjecture by providing a constructive method for obtaining a minimal $p$-encoder for any convolutional code over $\mathbb{Z}_{p^r}$.

cs.IT↗

arXiv2016

The dual of convolutional codes over $\mathbb{Z}_{p^r}$

An important class of codes widely used in applications is the class of convolutional codes. Most of the literature of convolutional codes is devoted to con- volutional codes over finite fields. The extension of the concept of convolutional codes from finite fields to finite rings have attracted much attention in recent years due to fact that they are the most appropriate codes for phase modulation. However convolutional codes over finite rings are more involved and not fully understood. Many results and features that are well-known for convolutional codes over finite fields have not been fully investigated in the context of finite rings. In this paper we focus in one of these unexplored areas, namely, we investigate the dual codes of convolutional codes over finite rings. In particular we study the p-dimension of the dual code of a convolutional code over a finite ring. This contribution can be considered a generalization and an extension, to the rings case, of the work done by Forney and McEliece on the dimension of the dual code of a convolutional code over a finite field.

math.RA↗