# On period polynomials of degree $2^m$ and weight distributions of certain irreducible cyclic codes

## TL;DR

This paper explicitly computes reduced cyclotomic periods of order 2^m over finite fields with specific characteristics, uses these to factor period polynomials, and determines weight distributions of related irreducible cyclic codes.

## Contribution

It provides explicit formulas for reduced cyclotomic periods and polynomials, and applies these to determine weight distributions of certain irreducible cyclic codes.

## Key findings

- Explicit values of reduced cyclotomic periods for order 2^m
- Factorizations of reduced period polynomials
- Descriptions of weight distributions of specific cyclic codes

## Abstract

We explicitly determine the values of reduced cyclotomic periods of order $2^m$, $m\ge 4$, for finite fields of characteristic $p\equiv 3$ or $5\pmod{8}$. These evaluations are applied to obtain explicit factorizations of the corresponding reduced period polynomials. As another application, the weight distributions of certain irreducible cyclic codes are described.