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

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

Problem Statement
The paper addresses the arithmetic characterization of integers $d$ that divide terms of the sequence $(a^{ks+T} + b^{ks+T})_{s \ge 1}$, where $a$ and $b$ are nonzero coprime integers, and $k, T$ are integers with $0 \le T < k$. These integers are termed $(T, k)$-good integers with respect to $(a, b)$. This work generalizes the classical notion of "good integers" (divisors of $a^s + b^s$), as well as "oddly-good" and "evenly-good" integers, which have previously served as critical arithmetic tools in the study of Euclidean and Hermitian dualities for cyclic and abelian codes. The primary challenge is to develop a self-contained arithmetic theory for these $(T, k)$-good integers and to apply this theory to the characterization of Galois dualities in coding theory, specifically for cyclic codes over finite fields $\mathbb{F}_{q^k}$.

Methodology
The authors employ a combination of elementary number theory and algebraic coding theory:

1. Arithmetic Analysis: The study begins by translating the divisibility condition $d \mid (a^{ks+T} + b^{ks+T})$ into congruences involving the multiplicative order of $ab^{-1}$ modulo $d$. The analysis is decomposed into:
   • Local Criteria: Characterization at odd prime powers $p^e$ using $p$-adic valuations and properties of multiplicative orders.
   • Global Criteria for Odd Integers: Combining local results via the Chinese Remainder Theorem to establish conditions based on the uniformity of 2-adic valuations of multiplicative orders across prime factors.
   • Even Case: A separate treatment for even integers, distinguishing between the case where exactly one of $a, b$ is even (yielding no even good integers) and where both are odd, relying on the 2-adic valuation of $a+b$.
   • Algorithmic Construction: An explicit algorithm is developed to decide membership in the set of $(T, k)$-good integers and to compute a corresponding exponent $s$.
2. Coding Theory Application: The arithmetic results are specialized to the case $(a, b) = (q, 1)$, where $q$ is a prime power. This links the arithmetic sets to $q^k$-cyclotomic classes on finite abelian groups. The stability of these classes under the map $g \mapsto -q^T g$ is shown to correspond exactly to the membership of the class order in the set of $(T, k)$-good integers.
3. Factorization and Duality: The theory is applied to the factorization of $x^n - 1$ over $\mathbb{F}_{q^k}$. The authors analyze $\theta$-self-reciprocal irreducible factors (where $\theta(\alpha) = \alpha^{q^T}$) and use these to characterize Galois Linear Complementary Dual (LCD) and Galois self-dual cyclic codes, including those with repeated roots.

Key Contributions and Results

• Arithmetic Theory of $(T, k)$-Good Integers:

  • Theorem 5 & Corollary 6: Provide necessary and sufficient conditions for an odd prime power $p^e$ to be $(T, k)$-good, expressed in terms of the multiplicative order of $ab^{-1}$ modulo $p^e$ and congruence conditions on $T$.
  • Theorem 9: Establishes a global characterization for odd integers $d$, requiring that the 2-adic valuation of the multiplicative order of $ab^{-1}$ is constant for all prime factors of $d$, and satisfying specific congruence and divisibility conditions involving $T$ and $k$.
  • Propositions 10 & 11: Characterize the even case, showing that for odd $a, b$, $2^\varepsilon d_0$ is $(T, k)$-good if and only if $d_0$ is $(T, k)$-good and specific conditions on $\varepsilon$ and the parity of $ks+T$ are met.
  • Algorithm 1: A practical procedure to determine membership in $G_{(T,k)}(a, b)$ and construct the exponent $s$.
  • Relations: The paper investigates the relationships between sets $G_{(T,k)}(a, b)$ as $T$ varies, introducing the set $C_k(T)$ to describe how membership propagates among different $T$ values (e.g., $G_{(T,k)} = G_{(k-T,k)}$ for odd integers).

• Applications to Coding Theory:

  • Cyclotomic Class Characterization (Theorem 23): A $q^k$-cyclotomic class $S_{q^k}(g)$ in a finite abelian group is of "type $T$" (stable under $g \mapsto -q^T g$) if and only if the order of $g$ belongs to $G_{(T,k)}(q, 1)$.
  • Enumeration (Proposition 26): Formulas are derived to count the number of $q^k$-cyclotomic classes of type $T$ and type $T'$ in finite abelian groups.
  • $\theta$-Self-Reciprocal Factors (Theorem 27): An irreducible factor $f_Q(x)$ of $x^n - 1$ is $\theta$-self-reciprocal if and only if the corresponding cyclotomic class $Q$ is of type $T$.
  • Galois LCD Codes (Theorem 30): A complete characterization of Galois LCD cyclic codes of length $n = n_0 p^r$ is provided. The generator polynomial is constructed by selecting $\theta$-self-reciprocal factors (with multiplicity 0 or $p^r$) and orbit products of non-self-reciprocal factors (with multiplicity 0 or $p^r$). An enumeration formula $2^{|\Omega_1| + |\Omega_2|}$ is given.
  • Galois Self-Dual Codes (Theorem 32 & Corollary 33): The paper proves that Galois self-dual cyclic codes exist only when $q$ and $n$ are even (repeated-root case). It provides explicit generator polynomial forms and enumeration formulas, distinguishing between general orbit structures and the involutory case (where orbits have size 2).

Significance and Claims
The paper claims to provide a unified arithmetic framework that extends the classical theory of good integers to the broader context of Galois duality. By introducing $(T, k)$-good integers, the authors bridge the gap between Euclidean ($T=0, k=1$) and Hermitian ($T=1, k=2$) dualities, encompassing them as special instances of a more general Galois duality defined by the automorphism $\theta(\alpha) = \alpha^{q^T}$.

The significance lies in the transfer of arithmetic properties of these integers to the structural analysis of cyclic codes. The results offer:

1. Explicit Criteria: Clear, computable conditions for determining the existence and structure of Galois LCD and self-dual codes.
2. Enumeration: Exact counting formulas for these code families based on the arithmetic properties of $G_{(T,k)}(q, 1)$.
3. Generalization: The framework is formulated for finite abelian groups, suggesting applicability beyond cyclic codes to abelian codes in group algebras.

The authors note that while the current work establishes the general theory and enumeration, future research could focus on specific code lengths and classes to derive more explicit structural descriptions and sharper formulas. The paper does not claim to solve specific open problems regarding specific code constructions but rather provides the theoretical machinery to analyze them.