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

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.