Expanding self-orthogonal codes over a ring $\Z_4$ to self-dual codes and unimodular lattices
<|reference_start|>Expanding self-orthogonal codes over a ring $\Z_4$ to self-dual codes and unimodular lattices: 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.<|reference_end|>
@article{shi2024expanding,
    title={Expanding self-orthogonal codes over a ring $\Z_4$ to self-dual codes
  and unimodular lattices},
    author={Minjia Shi, Sihui Tao, Jihoon Hong, Jon-Lark Kim},
    journal={arXiv preprint arXiv:2409.00404},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.00404},
    primaryClass={cs.CR}
}