Self-Conjugate-Reciprocal Irreducible Monic Factors of $x^n-1$ over Finite Fields and Their Applications

TL;DR

This paper characterizes and enumerates self-conjugate-reciprocal irreducible monic factors of $x^n-1$ over finite fields of square order, and explores their applications in coding theory, including Hermitian dual and self-dual codes.

Contribution

It provides a complete characterization, enumeration formulas, and recursive formulas for SCRIM factors over finite fields of square order, with applications to coding theory.

Findings

Derived characterization and enumeration formulas for SCRIM factors.

Established recursive formulas for counting SCRIM factors.

Applied results to Hermitian dual and self-dual cyclic codes.

Abstract

Self-reciprocal and self-conjugate-reciprocal polynomials over finite fields have been of interest due to their rich algebraic structures and wide applications. Self-reciprocal irreducible monic factors of $x^n-1$ over finite fields and their applications have been quite well studied. In this paper, self-conjugate-reciprocal irreducible monic (SCRIM) factors of $x^n-1$ over finite fields of square order have been focused on. The characterization of such factors is given together the enumeration formula. In many cases, recursive formulas for the number of SCRIM factors of $x^n-1$ have been given as well. As applications, Hermitian complementary dual codes over finite fields and Hermitian self-dual cyclic codes over finite chain rings of prime characteristic have been discussed.