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

Abstract: 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 $0\leq T&lt;k$, 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 $d\mid \bigl(a<sup>{ks+T}+b<sup>{ks+T}\bigr)$. Applications in coding theory are then obtained from the specialization $(a,b)=(q,1)$, where $q$ is a prime power. In particular, the $q<sup>k$-cyclotomic classes of the cyclic group $\mathbb Z_n$ characterize the Galois self-reciprocal irreducible factors of $x<sup>n-1$ over $\F_{q<sup>k}$, give a description and enumeration of Galois LCD cyclic codes of length $n$ over $\F_{q<sup>k}$, and lead to a characterization of Galois self-dual cyclic codes.

• The paper introduces (T,k)-good integers as a unified framework extending good, oddly-good, and evenly-good integers, then characterizes membership through multiplicative orders, congruences, and 2-adic conditions.
• The paper provides an explicit factorization-and-CRT algorithm that decides membership in G_(T,k)(a,b), constructs a valid exponent, and explains propagation relations among parameter values, especially for odd integer parts.
• The paper connects these arithmetic sets to q^k-cyclotomic classes and Galois duality, enabling criteria and enumeration formulas for self-reciprocal factors, Galois LCD codes, and repeated-root self-dual cyclic codes.

Motivation and the $(T,k)$-good integer framework

The paper by Jitman and Boonsuriyatham introduces a parametric extension of the classical notion of good integers. Recall that, for nonzero coprime integers $a$ and $b$, a positive integer $d$ is good with respect to $(a,b)$ if $d \mid (a^s + b^s)$ for some $s \ge 1$; the oddly-good and evenly-good variants restrict $s$ to be odd or even (2605.27933). The authors define $d$ to be a $(T,k)$-good integer with respect to $a$0 if

$a$1

for some positive integer $a$2, where $a$3. The set of all such integers is denoted $a$4. This family strictly generalizes the prior notions: $a$5, $a$6, and $a$7. A basic observation is that every element of $a$8 is coprime to $a$9, since $b$0.

The motivation is coding-theoretic: for $b$1, divisibility conditions of this form govern the reciprocal behavior of irreducible factors of $b$2 over finite fields, which in turn controls Euclidean, Hermitian, and—via the type-$b$3 action $b$4—Galois dualities of cyclic codes. Galois duality subsumes the Euclidean ($b$5, $b$6) and Hermitian ($b$7, $b$8) cases as special instances.

Arithmetic characterization

The technical core reduces the divisibility condition to multiplicative orders of $b$9 modulo divisors of $d$0. The key lemma states that $d$1 holds if and only if $d$2 is even and $d$3. Combined with order-lifting at odd prime powers (which preserves the $d$4-adic valuation of the order), this yields:

• Local criterion: an odd prime power $d$5 lies in $d$6 if and only if $d$7 is even and

$d$8

• Global criterion for odd integers: writing $d$9, membership holds if and only if there exists a positive integer $(a,b)$0 such that $(a,b)$1 for every prime $(a,b)$2, together with a congruence condition $(a,b)$3 and a divisibility condition involving $(a,b)$4. The proof proceeds via pairwise compatibility of congruence systems resolved by the Chinese Remainder Theorem.
• Even case: when $(a,b)$5 are both odd, $(a,b)$6 equals $(a,b)$7 for odd $(a,b)$8 and equals $(a,b)$9 for even $d \mid (a^s + b^s)$0. Consequently, $d \mid (a^s + b^s)$1 requires $d \mid (a^s + b^s)$2 and that $d \mid (a^s + b^s)$3 be odd for a suitable exponent; notably, $d \mid (a^s + b^s)$4 belongs whenever $d \mid (a^s + b^s)$5 does.

Two special cases simplify the criterion considerably: when $d \mid (a^s + b^s)$6 is odd, the $d \mid (a^s + b^s)$7-adic congruence condition vanishes entirely; when $d \mid (a^s + b^s)$8 is a power of two, the odd-part divisibility condition disappears, and for $d \mid (a^s + b^s)$9 with $s \ge 1$0, membership is characterized solely by $s \ge 1$1 across all primes dividing $s \ge 1$2.

The paper also establishes propagation relations among the sets via the subset $s \ge 1$3 of $s \ge 1$4: for odd $s \ge 1$5, membership in $s \ge 1$6 is equivalent to membership in $s \ge 1$7 for every $s \ge 1$8. In particular, $s \ge 1$9, and for odd prime $s$0,

$s$1

The authors explicitly note that these relations are stated only for the odd parts and do not automatically extend to full sets, because even-integer membership depends on the parity of $s$2.

Algorithmic decidability

The characterizations are assembled into an explicit algorithm that decides whether a given $s$3 belongs to $s$4 and, when it does, constructs an exponent $s$5 satisfying $s$6. The procedure factors the odd part of $s$7, checks local solvability of the linear congruence $s$8 at each prime power, combines solutions via CRT, and handles the $s$9-power part using $d$0 and the parity constraint on $d$1. Worked examples with $d$2 and $d$3 illustrate both positive and negative outcomes—for instance, $d$4 but $d$5—and tabulate the sets $d$6 for all six values of $d$7.

Cyclotomic classes of type $d$8

Specializing to $d$9, the paper studies $(T,k)$0-cyclotomic classes $(T,k)$1 in a finite abelian group $(T,k)$2 with $(T,k)$3. A class is of type $(T,k)$4 if it is stable under $(T,k)$5. The central structural result is that $(T,k)$6 is of type $(T,k)$7 if and only if $(T,k)$8; hence the type depends only on element order. Moreover, type-$(T,k)$9 stability propagates along $a$00: a class is of type $a$01 exactly when it is of type $a$02 for every $a$03, yielding in particular the symmetry between types $a$04 and $a$05.

This yields enumeration formulas: the number of classes of type $a$06 is

$a$07

where $a$08 is the exponent of $a$09 and $a$10 counts elements of order $a$11 (equal to $a$12 in the cyclic case). Concrete computations for $a$13 with $a$14, $a$15 show, e.g., $a$16 versus $a$17, illustrating how strongly the counts depend on $a$18.

Applications to Galois duality of cyclic codes

For $a$19, each $a$20-cyclotomic class $a$21 yields a monic irreducible factor $a$22 of $a$23 over $a$24. With $a$25, the paper proves

$a$26

so $a$27 is $a$28-self-reciprocal precisely when $a$29, i.e., when the elements of $a$30 have orders in $a$31. Partitioning cyclotomic classes into orbits under $a$32 organizes the factorization of $a$33: fixed points give self-reciprocal factors, while longer orbits give non-self-reciprocal blocks. An example over $a$34 with $a$35 shows that for $a$36 all eight quadratic factors are self-reciprocal, whereas for $a$37 only $a$38 is.

For repeated-root lengths $a$39, the paper characterizes Galois LCD cyclic codes: $a$40 is $a$41-LCD if and only if each self-reciprocal factor and each orbit product occurs in $a$42 with multiplicity either $a$43 or $a$44, giving exactly $a$45 such codes. For length $a$46 over $a$47 with $a$48, this yields $a$49 codes.

Regarding Galois self-dual cyclic codes, the paper proves a necessary-condition result with a clear scope restriction: such codes exist only when both $a$50 and $a$51 are even, so simple-root $a$52-self-dual cyclic codes do not exist. In the repeated-root case with $a$53 and $a$54, the generator polynomial must satisfy $a$55 on fixed-point classes and complementary exponents $a$56 on longer orbits. When the orbit sizes are even, exponents alternate between two complementary values; when an orbit has odd size, all its exponents are forced to equal $a$57. In the involutory case (all nontrivial orbits of size $a$58), the count simplifies to $a$59; for length $a$60 over $a$61 with $a$62, this gives $a$63 self-dual codes. The example with $a$64, where orbits have size $a$65, demonstrates that the involutory corollary does not apply and the orbit-wise theorem must be used instead—a point the authors handle correctly rather than forcing the simpler formula.

Limitations and open questions

The paper is candid about several boundaries of its results. The relations among sets $a$66 are established only for odd integers, and the authors state plainly that they do not extend automatically to even integers due to the parity dependence of the $a$67-adic valuation. The coding-theoretic applications are developed at the level of general theory; the authors note that deriving sharper structural descriptions and refined enumeration for specific code lengths and code families remains open. The abelian-group formulation suggests—but does not carry out—an extension to abelian codes in group algebras $a$68, and analogous treatments of constacyclic, quasi-abelian, and other Galois dual code families are left unaddressed. Additionally, the self-dual classification applies only in characteristic two with repeated roots; no analogue exists in the simple-root or odd-characteristic settings within this framework.

Conclusion

The paper develops a coherent arithmetic theory of $a$69-good integers that uniformly extends good, oddly-good, and evenly-good integers, complete with local and global characterizations, an explicit decision algorithm, and propagation relations among parameter values. Its specialization to $a$70 connects this arithmetic to $a$71-cyclotomic classes of type $a$72, yielding criteria for $a$73-self-reciprocal irreducible factors of $a$74 over $a$75 and explicit descriptions and enumerations of Galois LCD and Galois self-dual cyclic codes, including the repeated-root regime. The framework positions Galois duality as a common generalization of Euclidean and Hermitian duality controlled by a single arithmetic set, and identifies abelian codes and specific-length analyses as natural next targets.