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Good Integers: A Comprehensive Review with Applications

Somphong Jitman

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.

Good Integers: A Comprehensive Review with Applications

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 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.

Paper Structure

This paper contains 22 sections, 38 theorems, 43 equations, 4 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Good Integers
  3. Good Odd Integers
  4. Good Even Integers
  5. Useful Algorithm and Diagrams
  6. Oddly-Good and Evenly-Good Integers
  7. Oddly-Good Integers
  8. Evenly-Good Integers
  9. Use full Algorithm and Diagrams
  10. Applications
  11. Cyclic Codes and Factorization of $x^n-1$ over Finite Fields
  12. Factorization of $x^n-1$ over Finite Fields
  13. Cyclic Codes
  14. SRIM Factors of $x^n-1$ over Finite Fields and Applications
  15. SRIM Factors of $x^n-1$ over Finite Fields
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $a$ and $b$ be nonzero coprime integers and let $d$ be a positive integer. If $d\in G_{(a,b)}$, then $\gcd(a,d)=1=\gcd(b,d)$.

Figures (4)

  • Figure 1: Characterization of good integers with respect to coprime integers $a$ and $b$, where $ab$ is odd.
  • Figure 2: Characterization of good integers with respect to coprime integers $a$ and $b$, where $ab$ is even.
  • Figure 3: Characterization of oddly-good and evenly-good integers with respect to coprime integers $a$ and $b$, where $ab$ is even.
  • Figure 4: Characterization of oddly-good and evenly-good integers with respect to coprime integers $a$ and $b$, where $ab$ is even.

Theorems & Definitions (49)

  • Lemma 2.1: J2018a
  • Proposition 2.2: M1997
  • Proposition 2.3: M1997
  • Theorem 2.4: M1997
  • Example 2.1
  • Proposition 2.5: JPR2020
  • Proposition 2.6: J2018a
  • Proposition 2.7: JPR2020
  • Theorem 2.8
  • Example 2.2
  • ...and 39 more