Nonlinear Skew Quasi-Cyclic Codes

Abstract

This article explores nonlinear analogues of skew quasi-cyclic codes of index~$\ell$, 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 by using a two-fold iteration of the Smith normal form of matrices over skew polynomial rings. We show that actually a single use of the Smith normal form will suffice to determine the elementary divisors of the code. Along the way, we also describe duals of our codes with respect to appropriately chosen inner products.