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

TL;DR

This paper provides a comprehensive analysis and new methods for factorizing the polynomial x^n+1 over finite fields, with applications to negacyclic codes, offering explicit formulas and simplified code families.

Contribution

It introduces explicit and recursive factorization methods for x^n+1 over finite fields and revisits negacyclic codes with clearer, simpler forms.

Findings

New explicit factorization formulas for x^n+1

Recursive methods for polynomial factorization

Simplified negacyclic code families

Abstract

The polynomial $x^n+1$ over finite fields has been of interest due to its applications in the study of negacyclic codes over finite fields. In this paper, a rigorous treatment of the factorization of $x^n+1$ over finite fields is given as well as its applications. Explicit and recursive methods for factorizing $x^n+1$ over finite fields are provided together with the enumeration formula. As applications, some families of negacyclic codes are revisited with more clear and simpler forms.