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 $\mathbb {F}_{2}$ -vector space of $n$ -tuples with entries in $\mathbb {F}_{2}\oplus \mathbb {F} _{2}$ (the 2-dimensional vector space over $\mathbb {F}_{2}$ ) 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 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 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.