Optimal additive quaternary codes of low dimension

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 $k\leq 3.$ The most challenging case is dimension $k=2.5.$ We prove that an additive quaternary $[n,2.5,d]$-code where $d<n-1$ exists if and only if $3(n-d)\geq \lceil d/2\rceil +\lceil d/4\rceil +\lceil d/8\rceil$. 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 $3(m-e)\geq \lceil e/2\rceil +\lceil e/4\rceil+\lceil e/8\rceil$ is satisfied.Comment: 7 page

Additive quaternary codes related to exceptional linear quaternary codes

We study additive quaternary codes whose parameters are close to those of the extended cyclic $[{12,6,6}]_{4}$ -code or to the quaternary linear codes generated by the elliptic quadric in $PG(3,4)$ or its dual. In particular we characterize those codes in the category of additive codes and construct some additive codes whose parameters are better than those of any linear quaternary code. Our new code parameters are $[{22,17.5,4}]_{4}$