# Binary LCD Codes from $\mathbb{Z}_2\mathbb{Z}_2[u]$

## TL;DR

This paper extends the concept of LCD codes to a new ring structure, characterizes their properties, and constructs improved binary LCD codes from these ring-based codes.

## Contribution

It introduces $ ext{Z}_2 ext{Z}_2[u]$-LCD codes, characterizes their structure, and demonstrates how to generate binary LCD codes with better parameters.

## Key findings

- Binary images of $ ext{Z}_2 ext{Z}_2[u]$-LCD codes are binary LCD codes.
- New binary LCD codes with improved parameters are constructed.
- The paper generalizes LCD codes to a novel ring setting.

## Abstract

Linear complementary dual (LCD) codes over finite fields are linear codes satisfying $C\cap C^{\perp}=\{0\}$. We generalize the LCD codes over finite fields to $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD codes over the ring $\mathbf{Z}_2\times(\mathbf{Z}_2+u\mathbf{Z}_2)$. Under suitable conditions, $\mathbb{Z}_2\mathbb{Z}_2[u]$-linear codes that are $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD codes are characterized. We then prove that the binary image of a $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD code is a binary LCD code. Finally, by means of these conditions, we construct new binary LCD codes using $\mathbb{Z}_2\mathbb{Z}_2[u]$-LCD codes, most of which have better parameters than current binary LCD codes available.