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A Sequence Construction of Cyclic Codes over Finite Fields

Cunsheng Ding

TL;DR

The objective of this paper is to survey the progress in this direction in the past decade and to present some open problems.

Abstract

Cyclic codes over finite fields are widely implemented in data storage systems, communication systems, and consumer electronics, as they have very efficient encoding and decoding algorithms. They are also important in theory, as they are closely connected to several areas in mathematics. There are a few fundamental ways of constructing all cyclic codes over finite fields, including the generator matrix approach, the generator polynomial approach, the generating idempotent approach, and the $q$-polynomial approach. Another one is a sequence approach, which has been intensively investigated in the past decade. The objective of this paper is to survey the progress in the past decade in this direction.

A Sequence Construction of Cyclic Codes over Finite Fields

TL;DR

The objective of this paper is to survey the progress in this direction in the past decade and to present some open problems.

Abstract

Cyclic codes over finite fields are widely implemented in data storage systems, communication systems, and consumer electronics, as they have very efficient encoding and decoding algorithms. They are also important in theory, as they are closely connected to several areas in mathematics. There are a few fundamental ways of constructing all cyclic codes over finite fields, including the generator matrix approach, the generator polynomial approach, the generating idempotent approach, and the -polynomial approach. Another one is a sequence approach, which has been intensively investigated in the past decade. The objective of this paper is to survey the progress in the past decade in this direction.

Paper Structure

This paper contains 36 sections, 43 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some notation fixed throughout this paper
  4. Planar and APN polynomials
  5. The $q$-cyclotomic cosets modulo $n$
  6. The linear span and minimal polynomial of sequences
  7. Cyclic codes from combinatorial sequences
  8. The classical cyclic code ${\mathcal{C}}_{{\mathrm{GF}}(q)}(D)$ of a subset $D \in \mathbb{{Z}}_n$
  9. The cyclic code of the characteristic sequence of a subset $D \in \mathbb{{Z}}_n$
  10. Cyclic codes from a construction of sequences from polynomials over ${\mathrm{GF}}(q^m)$
  11. Cyclic codes from special monomials
  12. Binary cyclic codes from $f(x)=x^{2^{t}+3}$
  13. Binary cyclic codes from $f(x)=x^{2^h-1}$
  14. Binary cyclic codes from $f(x)=x^e$, $e=2^{(m-1)/2}+2^{(m-1)/4}-1$ and $m \equiv 1 \pmod{4}$
  15. Binary cyclic codes from $f(x)=x^{2^{2h}-2^h+1}$, where $\gcd(m,h)=1$
  16. ...and 21 more sections

Key Result

Lemma 1

Let $s^{\infty}$ be a sequence of period $L$ over ${\mathrm{GF}}(q)$. Define $S^{L}(x)= \sum_{i=0}^{L-1} s_i x^i \in {\mathrm{GF}}(q)[x].$ Then the minimal polynomial $\mathbb{M}_s(x)$ of $s^{\infty}$ is given by and the linear span ${\mathbb{L}}_s$ of $s^{\infty}$ is given by $L-\deg(\gcd(x^{L}-1, S^{L}(x))).$

Theorems & Definitions (78)

  • Lemma 1
  • Lemma 2
  • Example 3
  • Example 4
  • Theorem 6
  • Example 7
  • Example 8
  • Theorem 10
  • Example 11
  • Example 12
  • ...and 68 more