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<p>Cyclic codes, a significant subclass of linear codes, possess a well-defined algebraic structure and exhibit efficient encoding and decoding properties. These features facilitate their extensive application in data storage systems, communication systems, and consumer electronics. In this paper, through analyzing solutions to specific equations and irreducibility of polynomials over <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {F}}_{5^m}\)</EquationSource> </InlineEquation>, we construct five families of optimal quinary cyclic codes <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\mathcal {C}}_{(0,k,e)}\)</EquationSource> </InlineEquation> with parameters <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\([5^m-1, 5^m-2m-2, 4]\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(k=\frac{5^m+1}{2}\)</EquationSource> </InlineEquation>.</p>

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Several new classes of optimal quinary cyclic codes

  • 期刊论文

  • 发表日期：

    2025年12月26日

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  • Tingting Wu,
  • Lanqiang Li,
  • Li Liu

摘要

Cyclic codes, a significant subclass of linear codes, possess a well-defined algebraic structure and exhibit efficient encoding and decoding properties. These features facilitate their extensive application in data storage systems, communication systems, and consumer electronics. In this paper, through analyzing solutions to specific equations and irreducibility of polynomials over \({\mathbb {F}}_{5^m}\) , we construct five families of optimal quinary cyclic codes \({\mathcal {C}}_{(0,k,e)}\) with parameters \([5^m-1, 5^m-2m-2, 4]\) , where \(k=\frac{5^m+1}{2}\) .