Expanding self-orthogonal codes over a ring $\Z_4$ to self-dual codes and unimodular lattices
Minjia Shi, Sihui Tao, Jihoon Hong, Jon-Lark Kim
TL;DR
The paper studies expanding self-orthogonal codes over $\mathbb{Z}_4$ to self-dual codes over $\mathbb{Z}_4$ and connects the results to unimodular lattices via Construction $A$. It proves that any self-orthogonal $\mathbb{Z}_4$ code can be expanded to a self-dual one, with the possibility to preserve or increase the $k_1$ parameter and with $2k_1'+k_2'=n$. It reports five new Euclidean-optimal self-dual $\mathbb{Z}_4$ codes of lengths $27,28,29,33,34$ and constructs a new odd extremal unimodular lattice in dimension $34$ using Construction $A$, expanding known lattices and implications for design theory. The work provides practical algorithms for generating large families of $\mathbb{Z}_4$ self-dual codes and enriches the associated lattice theory with new examples and potential applications in quantum coding and secret sharing.
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.