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Selfdual skew cyclic codes

Xavier Caruso, Fabrice Drain

TL;DR

The paper addresses the problem of characterizing and enumerating selfdual skew cyclic codes within the Ore quotient ring $\mathbf{K}[X;\theta]/(X^{rk}-1)$. It develops an explicit bijection between such codes and products of isotropic subspaces in finite geometry, using an evaluation isomorphism that decomposes $\mathbf{E}_k$ into matrix algebras, and provides complete existence criteria, counting formulas, and efficient random generation methods in the separable case ($k$ coprime to the characteristic $p$). It also outlines an approach for the purely inseparable case ($k$ a power of $p$) via twisted separable codes and reports SageMath implementations for both separable and inseparable scenarios. The results yield practical algorithms for constructing and enumerating selfdual skew cyclic codes, with explicit formulas and a public SageMath package, enabling thorough exploration of their structure and applications. The work advances the understanding of duality in skew cyclic codes and connects coding theory to finite geometry through Segre-type and Witt-index techniques, with potential extensions to other constacyclic frameworks.

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.

Selfdual skew cyclic codes

TL;DR

The paper addresses the problem of characterizing and enumerating selfdual skew cyclic codes within the Ore quotient ring . It develops an explicit bijection between such codes and products of isotropic subspaces in finite geometry, using an evaluation isomorphism that decomposes into matrix algebras, and provides complete existence criteria, counting formulas, and efficient random generation methods in the separable case ( coprime to the characteristic ). It also outlines an approach for the purely inseparable case ( a power of ) via twisted separable codes and reports SageMath implementations for both separable and inseparable scenarios. The results yield practical algorithms for constructing and enumerating selfdual skew cyclic codes, with explicit formulas and a public SageMath package, enabling thorough exploration of their structure and applications. The work advances the understanding of duality in skew cyclic codes and connects coding theory to finite geometry through Segre-type and Witt-index techniques, with potential extensions to other constacyclic frameworks.

Abstract

Given a finite extension of degree of a finite field , we enumerate all selfdual skew cyclic codes in the Ore quotient ring for any positive integer coprime to the characteristic (separable case). We also provide an enumeration algorithm when is a power of (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 -linear subspaces of some extensions of . Finally, we report on an implementation in SageMath.

Paper Structure

This paper contains 27 sections, 22 theorems, 58 equations, 3 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Conventions and notations.
  4. Acknowledgments.
  5. From skew cyclic codes to finite geometry
  6. Definition of skew cyclic codes
  7. The evaluation isomorphism $\mathcal{E}_l$
  8. Adjunctions on $\mathbf{E}_{k}$ and related spaces
  9. Vector space duality
  10. Counting and generating selfdual skew cyclic codes
  11. Existence criterion
  12. Counting selfdual skew cyclic codes
  13. The nonpalindromic case
  14. The palindromic case
  15. Random generation of selfdual skew cyclic codes
  16. ...and 12 more sections

Key Result

Theorem 1.1

There exists an explicit bijection between the set of selfdual skew cyclic codes in $\mathbf{E}_{k}$ and the cartesian product of sets $W_{\mathtt{pal}}\times W_{\mathtt{nonpal}}$, where:

Figures (3)

  • Figure 1: Timings for the generation of one single code when $s=2$
  • Figure 2: Timings for the generation of one single code when $s=3$
  • Figure 3: Timings for the generation of one single code when $s=4$

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • ...and 52 more