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<p>In this work, we study linear codes with the folded Hamming distance, or equivalently, codes with the classical Hamming distance that are linear over a subfield. This includes additive codes. We study MDS codes in this setting and define quasi MDS (QMDS) codes and dually QMDS codes, which attain a more relaxed variant of the classical Singleton bound. We provide several general results concerning these codes, including restriction, shortening, weight distributions, existence, density, geometric description and bounds on their lengths relative to their field or alphabet sizes. We provide explicit examples and a binary construction with optimal lengths relative to their field or alphabet sizes, which beats any MDS code (in terms of length compared to the field or alphabet size).</p>

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Linear codes in the folded Hamming distance and the quasi MDS property

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月4日

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  • Umberto Martínez-Peñas,
  • Rubén Rodríguez-Ballesteros

摘要

In this work, we study linear codes with the folded Hamming distance, or equivalently, codes with the classical Hamming distance that are linear over a subfield. This includes additive codes. We study MDS codes in this setting and define quasi MDS (QMDS) codes and dually QMDS codes, which attain a more relaxed variant of the classical Singleton bound. We provide several general results concerning these codes, including restriction, shortening, weight distributions, existence, density, geometric description and bounds on their lengths relative to their field or alphabet sizes. We provide explicit examples and a binary construction with optimal lengths relative to their field or alphabet sizes, which beats any MDS code (in terms of length compared to the field or alphabet size).