Self-dual codes and quadratic residue codes over the ring $\mathbb{Z}_9+u\mathbb{Z}_9$

TL;DR

This paper introduces new Gray weight and Gray map definitions for codes over the ring Z_9+uZ_9, investigates self-dual codes, and explores quadratic residue codes, resulting in the construction of specific self-dual codes with given parameters.

Contribution

It presents novel definitions of Gray weight and Gray map for codes over Z_9+uZ_9, and constructs new self-dual codes with specific parameters.

Findings

Two self-dual codes with parameters [22,11,5] and [24,12,9] over Z_9 were obtained.

Structural properties of quadratic residue codes over the ring were analyzed.

New insights into self-dual code construction over Z_9+uZ_9 were provided.

Abstract

In this paper, we introduce a new definitions of the Gray weight and the Gray map for linear codes over $\mathbb{Z}_9+u\mathbb{Z}_9$ with $u^2=u$. Some results on self-dual codes over this ring are investigated. Further, the structural properties of quadratic residue codes are also considered. Two self-dual codes with parameters $[22,11,5]$ and $[24,12,9]$ over $\mathbb{Z}_9$ are obtained.