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<p>Self-dual codes have been studied actively because they are connected with mathematical structures including block designs and lattices and have practical applications in quantum error-correcting codes and secret sharing schemes. However, there has been relatively little attention given to constructing self-dual codes from self-orthogonal codes of smaller dimensions. Therefore, the main purpose of this paper is to propose a method to expand a self-orthogonal code over the ring <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {Z}_4\)</EquationSource> </InlineEquation> into one or more self-dual codes over <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {Z}_4\)</EquationSource> </InlineEquation>. We show that all self-dual codes over <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbb {Z}_4\)</EquationSource> </InlineEquation> of lengths 4 to 8 can be constructed in this manner. Furthermore, we have found five new self-dual codes over <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathbb {Z}_4\)</EquationSource> </InlineEquation> of lengths 27,&#xa0;28,&#xa0;29,&#xa0;33,&#xa0; and 34 with the highest Euclidean weight 12. Moreover, using Construction <i>A</i> applied to our new Euclidean-optimal self-dual codes over <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathbb {Z}_4\)</EquationSource> </InlineEquation>, we have constructed a new odd extremal unimodular lattice in dimension 34, whose kissing number was previously unknown.</p>

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Expanding self-orthogonal codes over a ring \(\mathbb {Z}_4\) to self-dual codes and unimodular lattices

  • 期刊论文

  • 发表日期：

    2026年5月1日

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  • Minjia Shi,
  • Sihui Tao

摘要

Self-dual codes have been studied actively because they are connected with mathematical structures including block designs and lattices and have practical applications in quantum error-correcting codes and secret sharing schemes. However, there has been relatively little attention given to constructing self-dual codes from self-orthogonal codes of smaller dimensions. Therefore, the main purpose of this paper is to propose a method to expand a self-orthogonal code over the ring \(\mathbb {Z}_4\) into one or more self-dual codes over \(\mathbb {Z}_4\) . We show that all self-dual codes over \(\mathbb {Z}_4\) of lengths 4 to 8 can be constructed in this manner. Furthermore, we have found five new self-dual codes over \(\mathbb {Z}_4\) of lengths 27, 28, 29, 33,  and 34 with the highest Euclidean weight 12. Moreover, using Construction A applied to our new Euclidean-optimal self-dual codes over \(\mathbb {Z}_4\) , we have constructed a new odd extremal unimodular lattice in dimension 34, whose kissing number was previously unknown.