One-Lee weight and two-Lee weight $\mathbb{Z}_2\mathbb{Z}_2[u]$-additive codes

TL;DR

This paper investigates specific weight properties of additive codes over a mixed algebraic structure, classifies certain self-dual codes, and derives optimal binary codes through Gray map transformations.

Contribution

It provides a complete classification of one-Lee weight self-dual codes and characterizes two-Lee weight projective codes over rac12;rac12;Z_2rac12;rac12;Z_2[u], introducing new code constructions.

Findings

Classified all one-Lee weight rac12;rac12;Z_2rac12;rac12;Z_2[u]-additive self-dual codes.

Determined the structure of two-Lee weight projective codes.

Derived optimal binary codes from these additive codes using Gray map.

Abstract

In this paper, we study one-Lee weight and two-Lee weight codes over $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$, where $u^{2}=0$. Some properties of one-Lee weight $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes are given, and a complete classification of one-Lee weight $\mathbb{Z}_2\mathbb{Z}_2[u]$-additive formally self-dual codes is obtained. The structure of two-Lee weight projective $\mathbb{Z}_2\mathbb{Z}_2[u]$ codes is determined. Some optimal binary linear codes are obtained directly from one-Lee weight and two-Lee weight $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes via the extended Gray map.