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<p>Maximum Distance Separable (MDS) codes are among the most fundamental objects in coding theory, and MDS matrices play a central role in their construction. In this paper, we investigate MDS matrices, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-orthogonal matrices, and several related matrix families for code design. Our contributions include multiple methods for constructing <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-orthogonal MDS matrices, a characterization of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-orthogonal Cauchy matrices, and techniques for obtaining linear complementary dual (LCD) codes, isodual codes, self-dual MDS codes, LCD MDS codes, and isodual MDS codes, not necessarily for even characteristics. Furthermore, we provide an explicit algorithm for constructing self-dual MDS codes over the field <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {F}_{{2^m}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>m</mi> </msup> </msub> </math></EquationSource> </InlineEquation>.</p>

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MDS matrices and their application to MDS and LCD code construction

  • 期刊论文

  • 发表日期：

    2026年5月2日

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  • D. Mokhtari,
  • K. Guenda,
  • T. A. Gulliver,
  • N. Aydin,
  • P. Liu

摘要

Maximum Distance Separable (MDS) codes are among the most fundamental objects in coding theory, and MDS matrices play a central role in their construction. In this paper, we investigate MDS matrices, \(\lambda \) λ -orthogonal matrices, and several related matrix families for code design. Our contributions include multiple methods for constructing \(\lambda \) λ -orthogonal MDS matrices, a characterization of \(\lambda \) λ -orthogonal Cauchy matrices, and techniques for obtaining linear complementary dual (LCD) codes, isodual codes, self-dual MDS codes, LCD MDS codes, and isodual MDS codes, not necessarily for even characteristics. Furthermore, we provide an explicit algorithm for constructing self-dual MDS codes over the field \(\mathbb {F}_{{2^m}}\) F 2 m .