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<p>In this paper, we extend the classification theory of self-dual matrix codes over finite fields due to Morrison to Galois rings. That is, we characterize a linear matrix-equivalence map defined as an <i>S</i>-automorphism of the module of matrices over Galois rings which preserves types of all matrices. This characterization has been already given by McDonald in the case for square matrices. In this paper, we present an alternative elementary proof of this result for matrices of arbitrary size using Morrison’s method. Also we characterize the subset of linear matrix-equivalence maps which commute with the operation taking the dual, and we define an equivalence of self-dual matrix codes by the maps. Finally using the equivalence, we classify self-dual matrix codes over <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({{\mathbb {Z}}}_4\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> with small sizes.</p>

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Self-dual matrix codes over Galois rings

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月20日

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  • Hijiri Kawazoe,
  • Makoto Tagami

摘要

In this paper, we extend the classification theory of self-dual matrix codes over finite fields due to Morrison to Galois rings. That is, we characterize a linear matrix-equivalence map defined as an S-automorphism of the module of matrices over Galois rings which preserves types of all matrices. This characterization has been already given by McDonald in the case for square matrices. In this paper, we present an alternative elementary proof of this result for matrices of arbitrary size using Morrison’s method. Also we characterize the subset of linear matrix-equivalence maps which commute with the operation taking the dual, and we define an equivalence of self-dual matrix codes by the maps. Finally using the equivalence, we classify self-dual matrix codes over \({{\mathbb {Z}}}_4\) Z 4 with small sizes.