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Four classes of optimal p-ary cyclic codes

Jinmei Fan, Jingyao Feng, Yuhan Men, Yanhai Zhang

TL;DR

The paper investigates construction of optimal $p$-ary cyclic codes with parameters $[p^m-1,p^m-2m-2,4]$ for odd primes $p\ge5$. It weakens weight-3 conditions and analyzes equations over finite fields to derive four constructions of codes of the form $C_p(0,1,u^{-1}v)$ or $C_p(0,1,uv^{-1})$ using $u=\frac{p^m+1}{2}$ and various $v$. Three of the four classes are infinite families and several known quinary codes are recovered as special cases. The work advances the theory of optimal cyclic codes by providing explicit algebraic criteria (weight-3 absence) and diverse exponent-based constructions.

Abstract

Let p>3 be an odd prime and m be a positive integer. Little progress on the study of optimal p-ary cyclic codes with parameters [p^m-1,p^m-2m-2,4] has been made.In this paper, by weakening the necessary and sufficient conditions on cyclic codes to have codewords of Hamming weight 3 and analyzing the solutions of certain equations over finite fields, four classes of optimal p-ary cyclic codes deduced by p^m+1/2 with parameters [p^m-1,p^m-2m-2,4] are presented.Wherein three classes of optimal p-ary cyclic codes are infinite.Many classes of known optimal quinary cyclic codes with parameters [5^m-1,5^m-2m-2,4] are special cases of the codes constructed in this paper.

Four classes of optimal p-ary cyclic codes

TL;DR

The paper investigates construction of optimal -ary cyclic codes with parameters for odd primes . It weakens weight-3 conditions and analyzes equations over finite fields to derive four constructions of codes of the form or using and various . Three of the four classes are infinite families and several known quinary codes are recovered as special cases. The work advances the theory of optimal cyclic codes by providing explicit algebraic criteria (weight-3 absence) and diverse exponent-based constructions.

Abstract

Let p>3 be an odd prime and m be a positive integer. Little progress on the study of optimal p-ary cyclic codes with parameters [p^m-1,p^m-2m-2,4] has been made.In this paper, by weakening the necessary and sufficient conditions on cyclic codes to have codewords of Hamming weight 3 and analyzing the solutions of certain equations over finite fields, four classes of optimal p-ary cyclic codes deduced by p^m+1/2 with parameters [p^m-1,p^m-2m-2,4] are presented.Wherein three classes of optimal p-ary cyclic codes are infinite.Many classes of known optimal quinary cyclic codes with parameters [5^m-1,5^m-2m-2,4] are special cases of the codes constructed in this paper.

Paper Structure

This paper contains 9 sections, 18 theorems, 36 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Four classes of new optimal $p$-ary cyclic codes with parameters $[p^m-1,p^m-2m-2,4]$
  4. The first class of optimal $p$-ary cyclic codes $C_p(0,1,u^{-1}v)$ or $C_p(0,1,uv^{-1})$ with parameters $[p^m-1,p^m-2m-2,4]$
  5. The second class of optimal $p$-ary cyclic codes $C_p(0,1,u^{-1}v)$ or $C_p(0,1,uv^{-1})$ with parameters $[p^m-1,p^m-2m-2,4]$
  6. The third class of optimal $p$-ary cyclic codes $C_p(0,1,u^{-1}v)$ or $C_p(0,1,uv^{-1})$ with parameters $[p^m-1,p^m-2m-2,4]$
  7. The fourth class of optimal $p$-ary cyclic codes $C_p(0,1,u^{-1}v)$ or $C_p(0,1,uv^{-1})$ with parameters $[p^m-1,p^m-2m-2,4]$
  8. Conclusions
  9. Acknowledgements

Key Result

Lemma 1

$($XCX$)$ Let $p$ be a prime and $m$ be a positive integer. For any $1\leq v\leq p^m-2$, then $|C_v|=m$ if ${\rm gcd}(v,p^m-1)< p$ or ${\rm gcd}(v,p^m-1){\rm gcd}(p^j-1,p^m-1)\not\equiv0\,({\rm mod}\,p^m-1)$ for all $1\leq j<m$.

Theorems & Definitions (40)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Theorem 1
  • proof
  • ...and 30 more