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.
Highlights
The polynomial xn + 1 over finite fields plays an important role in the study of negacyclic codes
A negacyclic code of length n over Fq can be uniquely determined by an ideal in the principal ring Fq[x]/〈xn + 1〉 generated by a monic divisor of xn + 1
We focus on the factorization of xn + 1 over finite fields Fq for arbitrary positive integers n and all odd prime powers 1ps q
Summary
The polynomial xn + 1 over finite fields plays an important role in the study of negacyclic codes (see [1,2,3,4,5] and references therein). In the case where the characteristic of F q is even, the factorization of xn + 1 xn − 1 over Fq has been given and applied in the study of cyclic codes over finite fields in [6]. Ordn′ (q) 1 over F is odd, a complete recursive q is provided together with factorization of a recursive formula for the number of its monic irreducible factors for all positive integers i. In the cases where ordn′ (q) is even, a recursive factorization of x2in′ + 1 over Fq is given for all positive integers i ≥ k.
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