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<p>Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regulargraphs. In this paper, by exploiting quadratic Gauss sums over finite fields, the parameters and weight distributions of six classes oflinear codes with at most six weights are entirely determined. More precisely, two classes of four weights linear codes, two classes offive weights linear codes, and two classes of six weights linear codes are derived. Furthermore, a class of optimal one weight codes andthree classes of optimal two weights codes which meet the Griesmer bound are given. The result from Magma shows that some optimalor almost optimal linear codes can be obtained from our construction. Moreover, some linear codes we constructed are minimal and canbe applied to secret sharing schemes based on their dual codes.</p>

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Several classes of linear codes with at most six weights and their secret sharing schemes

  • 期刊论文

  • 发表日期：

    2026年5月12日

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  • Qian Liu,
  • Xiaobei Dong,
  • Zhixiong Chen

摘要

Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regulargraphs. In this paper, by exploiting quadratic Gauss sums over finite fields, the parameters and weight distributions of six classes oflinear codes with at most six weights are entirely determined. More precisely, two classes of four weights linear codes, two classes offive weights linear codes, and two classes of six weights linear codes are derived. Furthermore, a class of optimal one weight codes andthree classes of optimal two weights codes which meet the Griesmer bound are given. The result from Magma shows that some optimalor almost optimal linear codes can be obtained from our construction. Moreover, some linear codes we constructed are minimal and canbe applied to secret sharing schemes based on their dual codes.