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

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.