Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $

Raj Kumar; Maheshanand Bhaintwal

doi:10.3934/amc.2020135

Article Contents

2023, Volume 17, Issue 2: 404-417. Doi: 10.3934/amc.2020135

This issue Previous Article Character sums over a non-chain ring and their applications Next Article The lower bounds on the second-order nonlinearity of three classes of Boolean functions

Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $

Raj Kumar and
Maheshanand Bhaintwal^,

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India

^* Corresponding author: Maheshanand Bhaintwal
^* Corresponding author: Maheshanand Bhaintwal

Received: August 2020

Early access: March 2021

Published: April 2023

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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 $.

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Mathematics Subject Classification: Primary: 94B05, 94B15.

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Table 1. Duadic codes of $ R(\mathbb{Z}_3\times \mathbb{Z}_3) $

$ \text{Duadic code}\; C $	$ \|C\| $	$ \psi(C) $	$ \phi(C) $

$ 2-1-0-1-0 $	$ 2^{18} $	$ [18, 4^82^2, 4 ]^* $	$ [18, 4^42^5, 8]^{**} $
$ u-1-0-1-0 $	$ 2^{18} $	$ [18, 4^92^0, 4 ] $	$ [18, 4^42^5, 6] $
$ (2+u)-1-0-1-0 $	$ 2^{18} $	$ [18, 4^92^0, 4 ] $	$ [18, 4^42^5, 6] $
$ 2-2-2-2-2 $	$ 2^{18} $	$ [18,4^02^{18},2 ]^* $	$ [18, 4^02^9, 4]^\dagger $
$ (2+u)-2-2-2-2 $	$ 2^{18} $	$ [18, 4^12^{16}, 4]^* $	$ [18, 4^02^9, 8]^{\dagger **} $
$ u-2-2-2-2 $	$ 2^{18} $	$ [18, 4^12^{16}, 4] $	$ [18, 4^02^9, 8]^{\dagger} $
$ 2-2-u-u-2 $	$ 2^{18} $	$ [18, 4^42^{10},4 ]^{**} $	$ [18, 4^02^9,8] $
$ u-2-u-u-2 $	$ 2^{18} $	$ [18, 4^52^8, 4]^{**} $	$ [18, 4^02^9,6]^\dagger $
$ (2+u)-2-u-u-2 $	$ 2^{18} $	$ [18,4^5 2^8, 4 ] $	$ [18, 4^02^9,8] $
$ (2+u) - (2+u) -0-1- (2+u) $	$ 2^{18} $	$ [18, 4^9 2^0, 4 ] $	$ [18, 4^22^5, 6]^{**} $
$ (2+u) - (2+u) -u-u- (2+u) $	$ 2^{18} $	$ [18, 4^9 2^0, 4 ] $	$ [18, 4^02^9,6] $
$ (2+u) - (2+u) -2-2- (2+u) $	$ 2^{18} $	$ [18, 4^9 2^0, 4 ] $	$ [18, 4^02^9,6] $
$ (2+u) - (2+u) -(2+u) - (2+u) - (2+u) $	$ 2^{18} $	$ [18, 4^9 2^0, 2 ] $	$ [18, 4^02^9,2]^\dagger $
$ u - (2+u) -2-2- (2+u) $	$ 2^{18} $	$ [18, 4^9 2^0, 4 ] $	$ [18, 4^02^9,8] $

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