Duadic codes over <inline-formula><tex-math id="M1">$ \mathbb{Z}_4+u\mathbb{Z}_4 $</tex-math></inline-formula>

Abstract

In this paper, we study the structure of duadic codes of an odd length $ n $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $, $ u^2 = 0 $, (more generally over $ \mathbb{Z}_{q}+u\mathbb{Z}_{q} $, $ u^2 = 0 $, where $ q = p^r $, $ p $ a prime and $ (n, p) = 1 $) using the discrete Fourier transform approach. We study these codes by considering them as a class of abelian codes. Some results related to self-duality and self-orthogonality of duadic codes are presented. Some conditions on the existence of self-dual augmented and extended duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ are determined. We present a sufficient condition for abelian codes of the same length over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ to have the same minimum Hamming distance. A new Gray map over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ is defined, and it is shown that the Gray image of an abelian code over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ is an abelian code over $ \mathbb{Z}_4 $. We have obtained five new linear codes of length $ 18 $ over $ \mathbb{Z}_4 $ from duadic codes of length $ 9 $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $ through the Gray map and a new map from $ \mathbb{Z}_4+u\mathbb{Z}_4 $ to $ \mathbb{Z}_4^2 $.