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<p>In this paper, we study quasi-abelian codes over a finite chain ring. First of all, we provide a decomposition of a group ring <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {R}G\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> through character theory for a given commutative group <i>G</i> and a finite commutative chain ring <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation>. Further, we characterize a special 2-quasi-abelian code over a finite chain ring <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> to be a linear complementary dual (LCD). Moreover, we give a method of construction of 2-quasi-abelian codes over a finite chain ring <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> from 2-quasi-abelian codes over a finite field <InlineEquation ID="IEq5"> <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>. Finally, we show that the class of LCD 2-quasi-abelian codes over a finite chain ring <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathcal {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> is asymptotically good.</p>

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Asymptotically good LCD 2-quasi-abelian codes over finite chain rings

  • 期刊论文

  • 发表日期：

    2025年3月7日

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  • Sanjit Bhowmick,
  • Xiusheng Liu

摘要

In this paper, we study quasi-abelian codes over a finite chain ring. First of all, we provide a decomposition of a group ring \(\mathcal {R}G\) R G through character theory for a given commutative group G and a finite commutative chain ring \(\mathcal {R}\) R . Further, we characterize a special 2-quasi-abelian code over a finite chain ring \(\mathcal {R}\) R to be a linear complementary dual (LCD). Moreover, we give a method of construction of 2-quasi-abelian codes over a finite chain ring \(\mathcal {R}\) R from 2-quasi-abelian codes over a finite field \(\mathbb {F}_q\) F q . Finally, we show that the class of LCD 2-quasi-abelian codes over a finite chain ring \(\mathcal {R}\) R is asymptotically good.