Expanding self-orthogonal codes over a ring $\Z_4$ to self-dual codes and unimodular lattices

• URL: http://arxiv.org/abs/2409.00404v1
• Date: Sat, 31 Aug 2024 09:38:42 GMT
• Title: Expanding self-orthogonal codes over a ring $\Z_4$ to self-dual codes and unimodular lattices
• Authors: Minjia Shi, Sihui Tao, Jihoon Hong, Jon-Lark Kim,
• Abstract summary: We show that all self-dual codes over $Z_4$ of lengths $4$ to $8$ can be constructed this way. We have found five new self-dual codes over $Z_4$ of lengths $27, 28, 29, 33,$ and $34$ with the highest Euclidean weight $12$.
• Score: 15.449427879628143
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: 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. Nevertheless, there has been less attention to construct self-dual codes from self-orthogonal codes with smaller dimensions. Hence, the main purpose of this paper is to propose a way to expand any self-orthogonal code over a ring $\Z_4$ to many self-dual codes over $\Z_4$. We show that all self-dual codes over $\Z_4$ of lengths $4$ to $8$ can be constructed this way. Furthermore, we have found five new self-dual codes over $\Z_4$ 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 $\Z_4$, we have constructed a new odd extremal unimodular lattice in dimension 34 whose kissing number was not previously known.

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