Classification of Optimal Linear ${\mathbb{Z}}_4$ Rate $1/2$ Codes of Length $\le 8$

Abstract

In this paper, we classify all optimal linear $[n,n/2]$ codes over ${\mathbb{Z}}_4$ up to length $n=8$, and determine the number of optimal codes which are self-dual and formally self-dual. Optimal codes with linear binary images are identified. In particular, we show that for length $8$, there are nine optimal codes for the Hamming distance, one optimal code for the Lee distance, and two optimal codes for the Euclidean distance.