Selfdual skew cyclic codes

TL;DR

This paper classifies and enumerates selfdual skew cyclic codes over finite field extensions, providing explicit formulas, algorithms for different cases, and an implementation in SageMath.

Contribution

It introduces a comprehensive enumeration method for selfdual skew cyclic codes over finite fields, including both separable and inseparable cases, with explicit bijections and algorithms.

Findings

Complete enumeration of selfdual skew cyclic codes in the separable case.

An enumeration algorithm for the inseparable case with some redundancies.

Implementation of the enumeration method in SageMath.

Abstract

Given a finite extension $K/F$ of degree $r$ of a finite field $F$, we enumerate all selfdual skew cyclic codes in the Ore quotient ring $K[X;\text{Frob}]/(X^{rk}-1)$ for any positive integer $k$ coprime to the characteristic $p$ (separable case). We also provide an enumeration algorithm when $k$ is a power of $p$ (purely inseparable case), at the cost of some redundancies. Our approach is based on an explicit bijection between skew cyclic codes, on the one hand, and certain families of $F$-linear subspaces of some extensions of $K$. Finally, we report on an implementation in SageMath.