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<p>Few-weight linear codes have important applications in the construction of strongly regular graphs, authentication codes and secret sharing schemes. In this paper, some few-weight linear codes are constructed from proper defining sets over finite fields. Their complete weight enumerators are explicitly determined using Weil sums. As applications, we give two classes of new projective three-weight linear codes, which achieve the Griesmer bound. We construct some new strongly regular graphs and infinite families of minimal three-weight linear codes with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_804_Article_IEq1.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(\frac{w_{min}}{w_{max}}\le \frac{p-1}{p}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <msub> <mi>w</mi> <mrow> <mi mathvariant="italic">min</mi> </mrow> </msub> <msub> <mi>w</mi> <mrow> <mi mathvariant="italic">max</mi> </mrow> </msub> </mfrac> <mo>≤</mo> <mfrac> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>p</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. Moreover, some new authentication codes are presented. Our results generalize and improve the work of Zhu and Liao (Finite Fields Appl. <b>75</b>, 101897, <CitationRef CitationID="CR44">2021</CitationRef>).</p>

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Complete weight enumerators of few-weight linear codes

  • 期刊论文

  • 发表日期：

    2025年5月15日

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  • Xiangdi Zeng,
  • Xiangrui Meng,
  • Jian Gao,
  • Fang-Wei Fu

摘要

Few-weight linear codes have important applications in the construction of strongly regular graphs, authentication codes and secret sharing schemes. In this paper, some few-weight linear codes are constructed from proper defining sets over finite fields. Their complete weight enumerators are explicitly determined using Weil sums. As applications, we give two classes of new projective three-weight linear codes, which achieve the Griesmer bound. We construct some new strongly regular graphs and infinite families of minimal three-weight linear codes with \(\frac{w_{min}}{w_{max}}\le \frac{p-1}{p}\) w min w max ≤ p - 1 p . Moreover, some new authentication codes are presented. Our results generalize and improve the work of Zhu and Liao (Finite Fields Appl. 75, 101897, 2021).