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<p>In recent studies, linear codes with few weights have been widely investigated for their utility in secret sharing schemes, authentication systems, association schemes, and strongly regular graphs. In this paper, we construct eight classes of three-weight linear codes, a class of four-weight linear codes, and a class of six-weight linear codes over finite fields by choosing suitable different defining sets. The parameters and weight distributions of these codes are completely determined by employing Weil sums. By using Magma computer experiments, we obtain many new (almost) optimal codes, some of which meet the Griesmer bound. Moreover, some linear codes we constructed are minimal and can be applied to secret sharing schemes based on their dual codes.</p>

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Constructions of several classes of linear codes from Weil sums and their secret sharing schemes

  • 期刊论文

  • 发表日期：

    2026年3月31日

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  • Qian Liu,
  • Jiafeng Song,
  • Zhizhu Lian

摘要

In recent studies, linear codes with few weights have been widely investigated for their utility in secret sharing schemes, authentication systems, association schemes, and strongly regular graphs. In this paper, we construct eight classes of three-weight linear codes, a class of four-weight linear codes, and a class of six-weight linear codes over finite fields by choosing suitable different defining sets. The parameters and weight distributions of these codes are completely determined by employing Weil sums. By using Magma computer experiments, we obtain many new (almost) optimal codes, some of which meet the Griesmer bound. Moreover, some linear codes we constructed are minimal and can be applied to secret sharing schemes based on their dual codes.