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

Li Zhu, Jinle Liu, Hongfeng Wu

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 $Φ_{n}(X)$ 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.

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

Abstract

Cyclotomic coset is a classical notion in the theory of finite field which has wide applications in various computation problems. Let be a prime power, and be a positive integer coprime to . In this paper we determine explicitly the representatives and the sizes of all -cyclotomic cosets modulo in the general settings. We introduce the definition of -adic cyclotomic system, which is a profinite space consists of certain compatible sequences of cyclotomic cosets. A precise characterization of the structure of the -adic cyclotomic system is given, which reveals the general formula for representatives of cyclotomic cosets. With the representatives and the sizes of -cyclotomic cosets modulo , we improve the formulas for the factorizations of and of over 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.

Paper Structure

This paper contains 15 sections, 30 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic number theory
  4. Finite fields and cyclic codes over finite fields
  5. A brief reminder on the multiple equal-difference representations of cyclotomic cosets
  6. Cyclic codes over finite fields
  7. Explicit representatives and sizes of cyclotomic cosets
  8. Primitive root systems
  9. $q$-cyclotomic cosets modulo an odd integer
  10. $q$-cyclotomic cosets modulo an even positive integer
  11. The classification of $q$-cyclotomic cosets modulo $n$
  12. Cyclotomic system
  13. The representatives and the sizes of $q$-cyclotomic cosets modulo $n$
  14. Irreducible factorizations of $X^{n}-1$ and of cyclotomic polynomials over $\mathbb{F}_{q}$
  15. Cyclic codes over finite fields

Key Result

Lemma 2.1

Nezami Let $\ell$ be an odd prime number, and $m$ be an integer such that $\ell \mid m-1$. Then $v_{\ell}(m^{d}-1) = v_{\ell}(m-1) + v_{\ell}(d)$ for any positive integer $d$.

Theorems & Definitions (54)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.1
  • Proposition 2.1
  • Theorem 2.1
  • Corollary 2.1
  • Theorem 2.2
  • Corollary 2.2
  • Definition 2.2
  • ...and 44 more