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Equivalence of constacyclic codes with shift constants of different orders

Reza Dastbasteh, Farzad Padashnick, Pedro M. Crespo, Markus Grassl, Javad Sharafi

Abstract

Let $a$ and $b$ be two non-zero elements of a finite field $\mathbb{F}_q$, where $q>2$. It has been shown that if $a$ and $b$ have the same multiplicative order in $\mathbb{F}_q$, then the families of $a$-constacyclic and $b$-constacyclic codes over $\mathbb{F}_q$ are monomially equivalent. In this paper, we investigate the monomial equivalence of $a$-constacyclic and $b$-constacyclic codes when $a$ and $b$ have distinct multiplicative orders. We present novel conditions for establishing monomial equivalence in such constacyclic codes, surpassing previous methods of determining monomially equivalent constacyclic and cyclic codes. As an application, we use these results to search for new linear codes more systematically. In particular, we present more than $70$ new record-breaking linear codes over various finite fields, as well as new binary quantum codes.

Equivalence of constacyclic codes with shift constants of different orders

Abstract

Let and be two non-zero elements of a finite field , where . It has been shown that if and have the same multiplicative order in , then the families of -constacyclic and -constacyclic codes over are monomially equivalent. In this paper, we investigate the monomial equivalence of -constacyclic and -constacyclic codes when and have distinct multiplicative orders. We present novel conditions for establishing monomial equivalence in such constacyclic codes, surpassing previous methods of determining monomially equivalent constacyclic and cyclic codes. As an application, we use these results to search for new linear codes more systematically. In particular, we present more than new record-breaking linear codes over various finite fields, as well as new binary quantum codes.
Paper Structure (5 sections, 8 theorems, 12 equations, 4 figures, 2 tables)

This paper contains 5 sections, 8 theorems, 12 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivalence of constacyclic codes with shift constants of different orders
  4. Classification of equivalent constacyclic codes over Small Finite Fields
  5. New record-breaking codes

Key Result

Theorem 2.3

Bograt Let $C_1$ and $C_2$ be two linear codes over a finite field. Then $C_1$ and $C_2$ are monomially equivalent if and only if they are isometric equivalent.

Figures (4)

  • Figure 1: Monomially equivalent $a$-constacyclic codes of length $n$ over $\mathbb{F}_3$ with $a\in \{1,2\}$.
  • Figure 2: Monomially equivalent $a$-constacyclic codes of length $n$ over $\mathbb{F}_4$ with $a\in \{1,\omega,\omega^2\}$.
  • Figure 3: Monomially equivalent $a$-constacyclic codes of length $n$ over $\mathbb{F}_5$ with $a\in \{1,2,3,4\}$.
  • Figure 4: Monomially equivalent $a$-constacyclic codes of length $n$ over $\mathbb{F}_7$ with $a\in \{1,2,\cdots, 6\}$.

Theorems & Definitions (26)

  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Example 3.3
  • ...and 16 more