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<p>NMDS codes and MDS codes have critical theoretical and practical value. In this paper, the authors develop a general construction of 3-dimensional NMDS codes of lengths from 2<sup><i>m</i></sup> to 2<sup><i>m</i></sup>+2 by selecting suitable generator matrices and determine their weight enumerators, where <i>m</i> ≥ 2 is an integer. In particular, the authors construct two types of 3-dimensional MDS codes and analyze the properties of the subfield codes of one of them. Then the authors derive some optimal locally recoverable codes via the NMDS codes. It is worth noting that all the NMDS and MDS codes are near Griesmer and Griesmer codes, respectively. Furthermore, the duals of the NMDS codes achieve length and dimension optimality, and of the MDS codes achieve distance optimality under the sphere packing bound. Finally, the authors use some of the codes constructed to build <i>s</i>-sum sets (where <i>s</i> &gt; 1 is odd), strongly regular graphs and 3-designs.</p>

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Constructions of Several Classes of Almost Optimal NMDS and Optimal MDS Codes

  • 期刊论文

  • 发表日期：

    2026年6月22日

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  • Xingbin Qiao,
  • Xiaoni Du,
  • Wenping Yuan

摘要

NMDS codes and MDS codes have critical theoretical and practical value. In this paper, the authors develop a general construction of 3-dimensional NMDS codes of lengths from 2m to 2m+2 by selecting suitable generator matrices and determine their weight enumerators, where m ≥ 2 is an integer. In particular, the authors construct two types of 3-dimensional MDS codes and analyze the properties of the subfield codes of one of them. Then the authors derive some optimal locally recoverable codes via the NMDS codes. It is worth noting that all the NMDS and MDS codes are near Griesmer and Griesmer codes, respectively. Furthermore, the duals of the NMDS codes achieve length and dimension optimality, and of the MDS codes achieve distance optimality under the sphere packing bound. Finally, the authors use some of the codes constructed to build s-sum sets (where s > 1 is odd), strongly regular graphs and 3-designs.