Optimal additive quaternary codes of low dimension

TL;DR

This paper determines the optimal parameters for low-dimensional additive quaternary codes, including the challenging 2.5-dimensional case, and constructs new optimal codes with specific parameters.

Contribution

It characterizes the existence of additive quaternary codes of dimension up to 3, especially the 2.5-dimensional case, and provides new optimal code constructions.

Findings

Optimal parameters for additive quaternary codes with dimension ≤ 3.

Existence conditions for 2.5-dimensional codes based on a new inequality.

New constructions of optimal 2.5-dimensional additive quaternary codes.

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 $k\leq 3.$ 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.