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<p>Binary cyclic codes having large dimensions and minimum distances close to the square-root bound are highly valuable in applications where high-rate transmission and robust error correction are both essential. They provide an optimal trade-off between these two factors, making them suitable for demanding communication and storage systems, post-quantum cryptography, radar and sonar systems, wireless sensor networks, and space communications. This paper aims to investigate cyclic codes by an efficient approach introduced by Ding [<CitationRef CitationID="CR5">5</CitationRef>] from several known classes of permutation monomials and trinomials over <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1661_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {F}}_{2^m}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>m</mi> </msup> </msub> </math></EquationSource> </InlineEquation>. We present several infinite families of binary cyclic codes of length <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1661_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <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> with dimensions larger than <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1661_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\((2^m-1)/2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mi>m</mi> </msup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. By applying the Hartmann-Tzeng bound, some of the lower bounds on the minimum distances of these cyclic codes are relatively close to the square root bound. Moreover, we obtain a new infinite family of optimal binary cyclic codes with parameters <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1661_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="172" /> </InlineMediaObject> <EquationSource Format="TEX">\([2^m-1,2^m-2-3m,8]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <msup> <mn>2</mn> <mi>m</mi> </msup> <mo>-</mo> <mn>1</mn> <mo>,</mo> <msup> <mn>2</mn> <mi>m</mi> </msup> <mo>-</mo> <mn>2</mn> <mo>-</mo> <mn>3</mn> <mi>m</mi> <mo>,</mo> <mn>8</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1661_Article_IEq7.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(m\ge 5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> is odd, according to the sphere-packing bound.</p>

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Binary cyclic codes from permutation polynomials over \({\mathbb {F}}_{2^m}\)

  • 期刊论文

  • 发表日期：

    2025年6月10日

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  • Mrinal Kanti Bose,
  • Udaya Parampalli,
  • Abhay Kumar Singh

摘要

Binary cyclic codes having large dimensions and minimum distances close to the square-root bound are highly valuable in applications where high-rate transmission and robust error correction are both essential. They provide an optimal trade-off between these two factors, making them suitable for demanding communication and storage systems, post-quantum cryptography, radar and sonar systems, wireless sensor networks, and space communications. This paper aims to investigate cyclic codes by an efficient approach introduced by Ding [5] from several known classes of permutation monomials and trinomials over \({\mathbb {F}}_{2^m}\) F 2 m . We present several infinite families of binary cyclic codes of length \(2^m-1\) 2 m - 1 with dimensions larger than \((2^m-1)/2\) ( 2 m - 1 ) / 2 . By applying the Hartmann-Tzeng bound, some of the lower bounds on the minimum distances of these cyclic codes are relatively close to the square root bound. Moreover, we obtain a new infinite family of optimal binary cyclic codes with parameters \([2^m-1,2^m-2-3m,8]\) [ 2 m - 1 , 2 m - 2 - 3 m , 8 ] , where \(m\ge 5\) m ≥ 5 is odd, according to the sphere-packing bound.