Expanding self-orthogonal codes over a ring 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 ₄ to many self-dual codes over ₄. We show that all self-dual codes over ₄ of lengths $4$ to $8$ can be constructed this way. Furthermore, we have found five new self-dual codes over 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 ₄, we have constructed a new odd extremal unimodular lattice in dimension 34 whose kissing number was not previously known.