Given a finite extension <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbf {K/F}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">K</mi> <mo stretchy="false">/</mo> <mi mathvariant="bold">F</mi> </mrow> </math></EquationSource> </InlineEquation> of degree r of a finite field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textbf{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">F</mi> </math></EquationSource> </InlineEquation>, we enumerate all selfdual skew cyclic codes in the Ore quotient ring <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textbf{E}_{k}:=\textbf{K}[X;\text {Frob}]/(X^{rk}-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">E</mi> <mi>k</mi> </msub> <mo>:</mo> <mo>=</mo> <mi mathvariant="bold">K</mi> <mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>;</mo> <mtext>Frob</mtext> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow> <mi mathvariant="italic">rk</mi> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textbf{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">F</mi> </math></EquationSource> </InlineEquation>-linear subspaces of some extensions of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textbf{K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">K</mi> </math></EquationSource> </InlineEquation>. Finally, we report on an implementation in SageMath.

Selfdual skew cyclic codes

期刊论文
发表日期：

2025年9月28日

查看全文

Xavier Caruso,
Fabrice Drain

摘要

Given a finite extension $\mathbf {K/F}$ $K / F$ of degree r of a finite field $\textbf{F}$ $F$ , we enumerate all selfdual skew cyclic codes in the Ore quotient ring $\textbf{E}_{k}:=\textbf{K}[X;\text {Frob}]/(X^{rk}-1)$ $E_{k} : = K [X; Frob] / (X^{rk} - 1)$ 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 $\textbf{F}$ $F$ -linear subspaces of some extensions of $\textbf{K}$ $K$ . Finally, we report on an implementation in SageMath.