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

TL;DR

This paper introduces a method to expand self-orthogonal codes over Z_4 into many self-dual codes, leading to new codes and the construction of an extremal unimodular lattice with previously unknown properties.

Contribution

It proposes a novel expansion technique for self-orthogonal codes over Z_4, enabling the construction of new self-dual codes and a new extremal unimodular lattice.

Findings

All self-dual codes over Z_4 of lengths 4 to 8 can be constructed using the method.

Five new self-dual codes over Z_4 of lengths 27, 28, 29, 33, and 34 were found.

A new odd extremal unimodular lattice in dimension 34 was constructed with an unknown kissing number.

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.