Good Integers: A Comprehensive Review with Applications

TL;DR

This paper thoroughly reviews the properties and applications of good integers, emphasizing their role in number theory and coding theory, especially in the construction of self-dual and complementary dual cyclic codes.

Contribution

It provides a systematic characterization, algorithms, and applications of good integers, bridging theoretical foundations with practical coding theory implications.

Findings

Characterization of good integers in number theory

Algorithms for computing and classifying good integers

Application in constructing self-dual cyclic codes

Abstract

For nonzero coprime integers $a$ and $b$, a positive integer $\ell$ is said to be \emph{good with respect to $a$ and $b$} if there exists a positive integer $k$ such that $\ell$ divides $a^{k} + b^{k}$. The concept of good integers has been the subject of continuous investigation since the 1990s due to their elegant number-theoretic properties and their significant applications in various areas, particularly in coding theory. This paper provides a comprehensive review of good integers, emphasizing both their theoretical foundations and their practical implications. We first revisit the fundamental number-theoretic properties of good integers and present their characterizations in a systematic manner. The exposition is enriched with well-structured algorithms and illustrative diagrams that facilitate their computation and classification. Subsequently, we explore applications of good integers in the study of algebraic coding theory. In particular, their roles in the characterization, construction, and enumeration of self-dual cyclic codes and complementary dual cyclic codes are discussed in detail. Several examples are provided to demonstrate the applicability of the theory. This review not only consolidates existing results but also highlights the unifying role of good integers in bridging number theory and coding theory.