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<p>For decades, the covering radius of cyclic codes over finite fields has been a topic of extensive research, owing to their wide applicability in areas such as data compression, testing, and write-once memory systems. However, most existing studies provide bounds for the covering radius of specific codes rather than determining their exact values, which is recognized as a challenging task. In this article, we focus on two specific classes of cyclic codes with lengths of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2^m - 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mi>m</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> defined over the prime field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {F}_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(m &gt; 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>&gt;</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> is a positive integer. We develop both algebraic and algorithmic methodologies based on systems of equations over finite fields. We employ elements in finite fields, such as trace functions, rational functions and additive characters, to calculate the second-order generalized covering radius value for these cyclic codes, offering new insights into their structural properties. Additionally, we explore the second-order generalized covering radius of the extended codes derived from one of these families of cyclic codes.</p>

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Determining the exact value of the second-order generalized covering radius of two classes of binary cyclic codes

  • 期刊论文

  • 发表日期：

    2026年3月21日

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  • Rong Luo,
  • Zhengchun Zhou,
  • Sihem Mesnager,
  • Vidya Sagar,
  • Haode Yan

摘要

For decades, the covering radius of cyclic codes over finite fields has been a topic of extensive research, owing to their wide applicability in areas such as data compression, testing, and write-once memory systems. However, most existing studies provide bounds for the covering radius of specific codes rather than determining their exact values, which is recognized as a challenging task. In this article, we focus on two specific classes of cyclic codes with lengths of \(2^m - 1\) 2 m - 1 defined over the prime field \(\mathbb {F}_2\) F 2 , where \(m > 3\) m > 3 is a positive integer. We develop both algebraic and algorithmic methodologies based on systems of equations over finite fields. We employ elements in finite fields, such as trace functions, rational functions and additive characters, to calculate the second-order generalized covering radius value for these cyclic codes, offering new insights into their structural properties. Additionally, we explore the second-order generalized covering radius of the extended codes derived from one of these families of cyclic codes.