Good Integers: A Comprehensive Review with Applications
Somphong Jitman

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 and , a positive integer is said to be \emph{good with respect to and } if there exists a positive integer such that divides . 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…
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