One-Sided Repeated-Root Two-Dimensional Cyclic and Constacyclic Codes

TL;DR

This paper explores the structure, duality, and asymptotic properties of one-sided repeated-root two-dimensional cyclic and constacyclic codes over finite fields, correcting previous inaccuracies and establishing new conditions.

Contribution

It provides generator matrices, polynomials, duality conditions, and links between asymptotic families of these codes, improving understanding and correcting prior results.

Findings

Derived generator matrices and polynomials for these codes.

Identified conditions for self-duality.

Established the existence of asymptotically good code families.

Abstract

In this paper, we study some repeated-root two-dimensional cyclic and constacyclic codes over a finite field $F=\mathbb{F}_q$. We obtain the generator matrices and generator polynomials of these codes and their duals. We also investigate when such codes are self-dual. Moreover, we prove that if there exists an asymptotically good family of one-sided repeated-root two-dimensional cyclic or constacyclic codes, then there exists an asymptotically good family of simple root two-dimensional cyclic or constacyclic codes with parameters at least as good as the first family. Furthermore, we show that several of the main results of the papers Rajabi and Khashyarmanesh (2018) and Sepasdar and Khashyarmanesh (2016) are not accurate and find other conditions needed for them to hold.