Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathbb {F}}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> be a finite field and G a finte group with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(|G|,q)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>G</mi> <mo stretchy="false">|</mo> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. By a group code in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathbb {F}}_q[G]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>G</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> we mean a two-sided ideal in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathbb {F}}_q[G]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>G</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We will prove a general criterion for the existence of group codes with given hull dimension, and then apply it to deduce explicit criterions for existence of group codes with hull dimension <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\le 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular our criterion for the existence of 1-dimensional hulls generalizes that of Luo et al. (Des Codes Cryptogr 92(12):4335–4352, 2024) which considers only abelian groups G.

On the hulls of group codes

期刊论文
发表日期：

2026年4月9日

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Xiheng Deng,
Yuan Ren

摘要

Let ${\mathbb {F}}_q$ $F_{q}$ be a finite field and G a finte group with $(|G|,q)=1$ $(| G |, q) = 1$ . By a group code in ${\mathbb {F}}_q[G]$ $F_{q} [G]$ we mean a two-sided ideal in ${\mathbb {F}}_q[G]$ $F_{q} [G]$ . We will prove a general criterion for the existence of group codes with given hull dimension, and then apply it to deduce explicit criterions for existence of group codes with hull dimension $\le 3$ $\leq 3$ . In particular our criterion for the existence of 1-dimensional hulls generalizes that of Luo et al. (Des Codes Cryptogr 92(12):4335–4352, 2024) which considers only abelian groups G.