Optimal additive quaternary codes of low dimension

Abstract

An additive quaternary $[n,k,d]$-code (length $n,$ quaternary dimension $k,$ minimum distance $d$) is a $2k$-dimensional F_2-vector space of $n$-tuples with entries in $Z_2\times Z_2$ (the $2$-dimensional vector space over F_2) with minimum Hamming distance $d.$ We determine the optimal parameters of additive quaternary codes of dimension $k\le 3.$ The most challenging case is dimension $k=\mathrm{2.5.}$ We prove that an additive quaternary $[n,2.5,d]$-code where $d<n-1$ exists if and only if $3(n-d)\ge \lceil d/2\rceil +\lceil d/4\rceil +\lceil d/8\rceil$. In particular we construct new optimal $2.5$-dimensional additive quaternary codes. As a by-product we give a direct proof for the fact that a binary linear $[3m,5,2e{]}_2$-code for $e<m-1$ exists if and only if the Griesmer bound $3(m-e)\ge \lceil e/2\rceil +\lceil e/4\rceil +\lceil e/8\rceil$ is satisfied.

7 pages