Nonlinear Reed-Solomon codes and nonlinear skew quasi-cyclic codes
Daniel Bossaller, Daniel Herden, Indalecio Ruiz-Bolanos
TL;DR
The paper addresses nonlinear generalizations of Reed-Solomon and skew quasi-cyclic codes by formulating them as modules over skew polynomial rings and quotient rings $R_n$ and $P_n$. It develops a robust Smith normal form framework, including a total-divisor theory, to classify the module structure and elementary divisors, sometimes requiring only a single SNF. By introducing $(\sigma,\ell)$-orbit vectors and a $q$-dual, it extends duality concepts to nonlinear skew codes and provides polynomial representations that model nonlinear skew QC codes as $P_n$-submodules of $R_n^\ell$. A principal contribution is the classification of nonlinear skew QC codes as left $P_n$-submodules, together with a dual structure via a $Q$-inner product and a $*$-dual that recovers the original code through double-duality under suitable bases. Overall, the work broadens the algebraic toolkit for nonlinear skew-cyclic codes, enabling systematic construction and analysis with potential practical impact in coding theory design.
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., $\mathbb{F}_{q^m}[x;σ]$-submodules of $\left(\mathbb{F}_{q^m}[x;σ]/(x^n - 1)\right)^\ell$. 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.