论文简报
math.NT 2605.27933v1 值得读

Good Integers: (T,k)-Subclasses and Applications to Galois Duality in Coding Theory

Somphong Jitman, Panthakan Boonsuriyatham

发布日期：2026-05-27 04:05 相关性：1.0000 价值：0.7800 分类：math.NT cs.IT
arXiv 摘要 PDF 原文 网页正文

摘要

The notion of good integers, namely the divisors of the sequence $(a^s+b^s)_{s\ge 1}$ for nonzero coprime integers $a$ and $b$, together with their subfamilies such as oddly-good and evenly-good integers, has become an important arithmetic tool in the study of Euclidean and Hermitian dualities for abelian and cyclic codes. Building on this perspective, this paper introduces and studies another interesting subclass of good integers arising from the sequence $\bigl(a^{ks+T}+b^{ks+T}\bigr)_{s\ge 1}$ for some integers $0\leq T<k$, whose divisors are called $(T,k)$-{\em good integers with respect to} $(a,b)$. An arithmetic theory of these integers is developed, including a characterization at odd prime powers, a general characterization for odd integers in terms of $2$-adic valuations, and a treatment of even integers. An explicit algorithm is also given for deciding whether a given integer $d$ is $(T,k)$-good with respect to $(a,b)$ and, when it is, for computing an exponent $s$ such that $d\mid \bigl(a^{ks+T}+b^{ks+T}\bigr)$. Applications in coding theory are then obtained from the specialization $(a,b)=(q,1)$, where $q$ is a prime power. In particular, the $q^k$-cyclotomic classes of the cyclic group $\mathbb Z_n$ characterize the Galois self-reciprocal irreducible factors of $x^n-1$ over $\F_{q^k}$, give a description and enumeration of Galois LCD cyclic codes of length $n$ over $\F_{q^k}$, and lead to a characterization of Galois self-dual cyclic codes.

相关性判断

high
相关方向
coding_theory information_theory galois_duality cyclic_codes
判断依据

The paper is directly in cs.IT and develops arithmetic tools with explicit applications to Galois duality, cyclic codes, LCD/self-dual codes, and cyclotomic classes, which are squarely in coding theory and adjacent information theory.

价值判断

Directly relevant to coding theory through Galois duality, cyclic codes, LCD cyclic codes, self-dual cyclic codes, and cyclotomic class structure. Structure analysis indicates substantial technical content: prime-power criteria, 2-adic/global characterizations, CRT-style local-to-global reasoning, and an explicit decision/construction algorithm. The contribution appears more like a rigorous arithmetic framework with coding-theory applications than an immediately broad information-theory breakthrough, so deep review is justified but not urgent must-read.

核心问题与主要方法

核心问题

Characterize divisibility by terms of a^{ks+T}+b^{ks+T} and use it to analyze Galois duality structures in cyclic codes

场景：Arithmetic over coprime integers a,b with 0<=T<k; specialized to q^k-cyclotomic classes and cyclic codes over F_{q^k}

主要方法

Reduces d | a^N+b^N to the condition that ord_d(ab^{-1}) is even and N is congruent to half that order modulo ord_d(ab^{-1}). For odd prime powers, solves the constrained exponent form N=ks+T via a linear congruence modulo ord_{p^e}(ab^{-1}). Combines prime-power constraints for odd d using a common 2-adic valuation alpha and Chinese Remainder Theorem compatibility. Handles even d separately through the odd part plus 2-adic divisibility of a^{ks+T}+b^{ks+T}, including parity of ks+T when a,b are odd. Transfers arithmetic membership in G_{(T,k)}(q,1) to orbit stability under g -> -q^T g on q^k-cyclotomic classes. Uses sigma-orbits of q^k-cyclotomic classes under Q -> -q^T Q to organize theta-reciprocal factorization and generator-polynomial descriptions for Galois LCD and self-dual cyclic codes.

关键贡献与后续阅读

关键贡献

Defines the parameterized class G_{(T,k)}(a,b), unifying classical good integers, evenly-good integers, and oddly-good integers through choices of T and k. Provides arithmetic membership criteria for (T,k)-good integers at odd prime powers, for general odd integers via 2-adic valuations, and for even integers via separate 2-adic restrictions. Gives an explicit decision and construction algorithm for determining d in G_{(T,k)}(a,b) and producing an exponent s when membership holds. Establishes that, for (a,b)=(q,1), the order of an element in a finite abelian group determines whether its q^k-cyclotomic class is stable under g -> -q^T g. Applies the orbit criterion to theta-self-reciprocal irreducible factors of x^n-1 over F_{q^k}. Derives generator-polynomial descriptions and enumeration formulas for Galois LCD cyclic codes and characterizes repeated-root Galois self-dual cyclic codes, including a simpler involutory-orbit case.

研究启发

How does the membership algorithm scale with large d, especially when factoring d or computing multiplicative orders dominates? Are the enumeration formulas computationally practical for code lengths beyond the illustrative examples? Which known Euclidean or Hermitian cyclic-code classification results are recovered exactly as special cases, and which are genuinely new in the Galois setting? Can the abelian-group formulation be turned into explicit abelian-code construction and enumeration results, as suggested in the conclusion?

限制与不确定性

Primary category is math.NT, so some value may be concentrated in arithmetic classification rather than coding-theory performance or constructions. Structure notes limited explicit examples and further work needed for sharper code-length results and broader code families. No full-paper review was used, so novelty and impact are inferred from relevance and structure only.

参考文献

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