Explicit Representatives and Sizes of Cyclotomic Cosets and their Application to Cyclic Codes over Finite Fields

TL;DR

This paper explicitly determines the representatives and sizes of cyclotomic cosets modulo n, introduces a 2-adic system, and applies these results to classify and enumerate cyclic codes over finite fields.

Contribution

It provides explicit formulas for cyclotomic cosets and applies these to improve factorization formulas and classify cyclic codes, including self-dual codes.

Findings

Explicit representatives and sizes for all q-cyclotomic cosets.

Enhanced formulas for factorization of X^n - 1 and cyclotomic polynomials.

Complete classification and enumeration of self-dual cyclic codes.

Abstract

Cyclotomic coset is a classical notion in the theory of finite field which has wide applications in various computation problems. Let $q$ be a prime power, and $n$ be a positive integer coprime to $q$. In this paper we determine explicitly the representatives and the sizes of all $q$-cyclotomic cosets modulo $n$ in the general settings. We introduce the definition of $2$-adic cyclotomic system, which is a profinite space consists of certain compatible sequences of cyclotomic cosets. A precise characterization of the structure of the $2$-adic cyclotomic system is given, which reveals the general formula for representatives of cyclotomic cosets. With the representatives and the sizes of $q$-cyclotomic cosets modulo $n$, we improve the formulas for the factorizations of $X^{n}-1$ and of $\Phi_{n}(X)$ over $\mathbb{F}_{q}$ given in \cite{Graner}. As a consequence, we classify the cyclic codes over finite fields via giving their generator polynomials. Moreover, the self-dual cyclic codes are determined and enumerated.