Nonlinear Reed-Solomon codes and nonlinear skew quasi-cyclic codes

TL;DR

This paper explores nonlinear generalizations of Reed-Solomon and skew quasi-cyclic codes, analyzing their structure using Smith normal form over skew polynomial rings and identifying conditions for elementary divisor determination.

Contribution

It introduces nonlinear skew quasi-cyclic codes and develops methods to analyze their module structure via Smith normal form in skew polynomial rings.

Findings

Nonlinear codes generalize Reed-Solomon codes.

Module structure of nonlinear skew quasi-cyclic codes characterized.

Single Smith normal form application can determine elementary divisors in some cases.

Abstract

This article begins with an exploration of nonlinear codes ($\mathbb{F}_q$-linear subspaces of $\mathbb{F}_{q^m}^n$) which are generalizations of the familiar Reed-Solomon codes. This then leads to a wider exploration of nonlinear analogues of the skew quasi-cyclic codes of index $\ell$ first explored in 2010 by Abualrub et al., i.e., $\mathbb{F}_{q^m}[x;\sigma]$-submodules of $\left(\mathbb{F}_{q^m}[x;\sigma]/(x^n - 1)\right)^\ell$. After introducing nonlinear skew quasi-cyclic codes, we then determine the module structure of these codes using a two-fold iteration of the Smith normal form of matrices over skew polynomial rings. Finally we show that in certain cases, a single use of the Smith normal form will suffice to determine the elementary divisors of the code.