An explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $

Yuan Cao; Yonglin Cao; Hai Q. Dinh; Ramakrishna Bandi; Fang-Wei Fu

doi:10.3934/amc.2020067

Article Contents

2021, Volume 15, Issue 2: 291-309. Doi: 10.3934/amc.2020067

This issue Previous Article Two classes of near-optimal codebooks with respect to the Welch bound Next Article Cryptographic properties of cyclic binary matrices

An explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $

1.
School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255091, China
2.
Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China
3.
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114, China
4.
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
5.
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
6.
Department of Mathematics, Dr. SPM IIIT Naya Raipur, Atal Nagar 493661, India
7.
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin Key Laboratory of Network and Data Security Technology, Tianjin 300071, China

^* Corresponding author: Yonglin Cao
^* Corresponding author: Yonglin Cao

Received: June 2019

Revised: October 2019

Early access: January 2020

Published: May 2021

This research is supported in part by National Natural Science Foundation of China (Grant Nos. 11801324, 11671235, 61971243, 61571243), the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2018BA007), the Scientific Research Foundation for the PhD of Shandong University of Technology (Grant No. 417037), the Scientific Research Fund of Hubei Provincial Key Laboratory of Applied Mathematics (Hubei University) (Grant Nos. HBAM201906, HBAM201804) and the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2018MMAEZD09) and the Nankai Zhide Foundation

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Abstract

In this paper, we give an explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over the local non-principal ideal ring $ R = \mathbb{Z}_4+u\mathbb{Z}_4 $ $ (u^2 = 0) $, where $ k, n $ are arbitrary positive integers and $ n $ is odd. In particular, we present all distinct negacyclic codes of length $ 2^k $ over $ R $ precisely. Moreover, we provide an exact mass formula for the number of negacyclic codes of length $ 2^kn $ over $ R $ and correct several mistakes in some literatures.

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Mathematics Subject Classification: Primary: 94B15, 94B05; Secondary: 11T71.

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