Some generalizations of good integers &nbsp;and &nbsp;their applications in the &nbsp;study &nbsp;of &nbsp;self-dual negacyclic codes

Somphong Jitman; Supawadee Prugsapitak; Madhu Raka

doi:10.3934/amc.2020004

Article Contents

2020, Volume 14, Issue 1: 35-51. Doi: 10.3934/amc.2020004

This issue Previous Article A construction of bent functions with optimal algebraic degree and large symmetric group Next Article Skew constacyclic codes over the local Frobenius non-chain rings of order 16

Some generalizations of good integers and their applications in the study of self-dual negacyclic codes

1.
Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand
2.
Algebra and Applications Research Unit, Department of Mathematics, and Statistics, Faculty of Science, Prince of Songkla University, Hatyai, Songkhla 90110, Thailand
3.
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-160014, India

^* Corresponding author
^* Corresponding author

Received: May 2018

Published: August 2019

S. Jitman was supported by the Thailand Research Fund and Silpakorn University under Research Grant RSA6280042. M. Raka was supported by the Council of Scientific and Industrial Research (CSIR), India, Sanction No. 21(1042)/17/EMR-II.

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Abstract

Good integers introduced in 1997 form an interesting family of integers that has been continuously studied due to their rich number theoretical properties and wide applications. In this paper, we have focused on classes of $ 2^\beta $-good integers, $ 2^\beta $-oddly-good integers, and $ 2^\beta $-evenly-good integers which are generalizations of good integers. Properties of such integers have been given as well as their applications in characterizing and enumerating self-dual negacyclic codes over finite fields. An alternative proof for the characterization of the existence of a self-dual negacyclic code over finite fields has been given in terms of such generalized good integers. A general enumeration formula for the number of self-dual negacyclic codes of length $ n $ over finite fields has been established. For some specific lengths, explicit formulas have been provided as well. Some known results on self-dual negacyclic codes over finite fields can be formalized and viewed as special cases of this work.

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Mathematics Subject Classification: Primary: 11N25, 94B15; Secondary: 94B60.

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