Some Generalizations of Good Integers and Their Applications in the Study of Self-Dual Negacyclic Codes

TL;DR

This paper generalizes the concept of good integers to classes like 2^β-good integers and explores their properties and applications in characterizing and enumerating self-dual negacyclic codes over finite fields.

Contribution

It introduces new classes of generalized good integers and applies them to characterize and count self-dual negacyclic codes, providing new proofs and explicit formulas.

Findings

Properties of generalized good integers are established.

An enumeration formula for self-dual negacyclic codes is derived.

Explicit formulas are provided for specific code lengths.

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.