SRIM and SCRIM factors of xn + 1 over finite fields and their applications

Abstract

Self-Reciprocal Irreducible Monic (SRIM) and Self-Conjugate-Reciprocal Irreducible Monic (SCRIM) factors of [Formula: see text] over finite fields have become of interest due to their rich algebraic structures and wide applications. In this paper, these notions are extended to factors of [Formula: see text] over finite fields. Characterization and enumeration of SRIM and SCRIM factors of [Formula: see text] over finite fields are established. Simplification and recessive formulas for the number of such factors are given. Finally, applications in the study of complementary negacyclic codes are discussed.

Revisiting the Factorization of x n + 1 over Finite Fields with Applications

“…Lemma 2 (see [10], Lemma 3). Let q be an odd prime power, and let n ′ be an odd positive integer such that gcd(q, n ′ ) � 1.…”

“…In [10], a basic idea for the factorization of x 2 i n′ + 1 is given using (3) and the following lemmas.…”

“…In general, negacyclic codes have been studied in [3,4,10]. Here, we focus on negacyclic codes of length n � p s 2 i n ′ with i ≥ k, where p is the characteristic of F q .…”

Revisiting the Factorization of $x^n+1$ over Finite Fields

“…In [4], a basic idea for the factorization of x 2 i n ′ + 1 is given using (2.1) and the following lemmas.…”

“…Let λ be the positive integer such that 2 λ ||ord n ′ (q) and let β be the positive integer such that 2 β ||(q 2 − 1). Let In general, negacyclic codes have been studied in [4], [5], and [9]. Here, we focus on negacyclic codes of length n = p s 2 i n ′ with i ≥ k, where p is the characteristic…”