Good Integers and Applications in Coding Theory

TL;DR

This paper provides a complete characterization of good integers, introduces subclasses, and explores their applications in coding theory, specifically in analyzing the hulls of abelian codes.

Contribution

It offers a full characterization of good integers, introduces subclasses, and applies these concepts to study hulls of abelian codes.

Findings

Complete characterization of all good integers

Introduction of oddly-good and evenly-good subclasses

Bounds on the average dimension of hulls of abelian codes

Abstract

A class of good integers has been introduced by P. Moree in $1997$ together with the characterization of good odd integers. Such integers have shown to have nice number theoretical properties and wide applications. In this paper, a complete characterization of all good integers is given. Two subclasses of good integers are introduced, namely, oddly-good and evenly-good integers. The characterization and properties of good integers in these two subclasses are determined. As applications, good integers and oddly-good integers are applied in the study of the hulls of abelian codes. The average dimension of the hulls of abelian codes is given together with some upper and lower bounds.