Combinatorial Constructions of Optimal Quaternary Additive Codes

TL;DR

This paper introduces new combinatorial methods to construct optimal quaternary additive codes with non-integer dimensions, expanding the possibilities for code design in this domain.

Contribution

It proposes combinatorial constructions and a generalized Construction X to build non-integer dimensional optimal additive codes from linear codes.

Findings

Constructed ten classes of optimal quaternary non-integer dimensional additive codes.

Determined optimal additive [n,3.5,n-t]_4 codes for most t values.

Developed new combinatorial and generalized construction methods.

Abstract

This paper aims to construct optimal quaternary additive codes with non-integer dimensions. Firstly, we propose combinatorial constructions of quaternary additive constant-weight codes, alongside additive generalized anticode construction. Subsequently, we propose generalized Construction X, which facilitates the construction of non-integer dimensional optimal additive codes from linear codes. Then, we construct ten classes of optimal quaternary non-integer dimensional additive codes through these two methods. As an application, we also determine the optimal additive $[n,3.5,n-t]_4$ codes for all $t$ with variable $n$, except for $t=6,7,12$.