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<p>Linear codes with few weights have been extensively developed because of their wide applications in consumer electronics, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper is devoted to two new constructions of linear codes with few weights over the ring <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {F}_p+u\mathbb {F}_p\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>p</mi> </msub> <mo>+</mo> <mi>u</mi> <msub> <mi mathvariant="double-struck">F</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> from projective spaces. Moreover, we determine the Lee weight distributions of these codes by investigating the property of the support of the vectors of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {F}_p^m\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">F</mi> <mi>p</mi> <mi>m</mi> </msubsup> </math></EquationSource> </InlineEquation>. Via the Gray map, we obtain three classes of linear codes with few weights over <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {F}_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>. In some cases, these linear codes are proved to be minimal and optimal with respect to the Griesmer bound.</p>

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Optimal few-weight codes from projective spaces

  • 期刊论文

  • 发表日期：

    2025年1月24日

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  • Guangkui Xu,
  • Gaojun Luo,
  • Heqian Xu,
  • Song Xu

摘要

Linear codes with few weights have been extensively developed because of their wide applications in consumer electronics, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper is devoted to two new constructions of linear codes with few weights over the ring \(\mathbb {F}_p+u\mathbb {F}_p\) F p + u F p from projective spaces. Moreover, we determine the Lee weight distributions of these codes by investigating the property of the support of the vectors of \(\mathbb {F}_p^m\) F p m . Via the Gray map, we obtain three classes of linear codes with few weights over \(\mathbb {F}_p\) F p . In some cases, these linear codes are proved to be minimal and optimal with respect to the Griesmer bound.