SRIM and SCRIM Factors of $x^n+1$ over Finite Fields and Their Applications

TL;DR

This paper extends the study of SRIM and SCRIM factors from $x^n-1$ to $x^n+1$ over finite fields, providing characterization, enumeration, and applications in negacyclic codes.

Contribution

It introduces the concepts of SRIM and SCRIM factors for $x^n+1$, offering new characterization, enumeration formulas, and exploring their applications in coding theory.

Findings

Characterization and enumeration formulas for SRIM and SCRIM factors of $x^n+1$

Simplified formulas for counting such factors

Applications in complementary negacyclic codes

Abstract

Self-Reciprocal Irreducible Monic (SRIM) and Self-Conjugate-Reciprocal Irreducible Monic (SRCIM) factors of $x^n-1$ 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 $x^n+ 1$ over finite fields. Characterization and enumeration of SRIM and SCRIM factors of $x^n+1$ over finite fields are established. Simplification and recessive formulas for the number of such factors are given. Finally, applications in the studied of complementary negacyclic codes are discussed.