Equivalence of constacyclic codes with shift constants of different orders
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912125554786304 |
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| author | Dastbasteh, Reza Padashnick, Farzad Crespo, Pedro M. Grassl, Markus Sharafi, Javad |
| author_facet | Dastbasteh, Reza Padashnick, Farzad Crespo, Pedro M. Grassl, Markus Sharafi, Javad |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04600 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivalence of constacyclic codes with shift constants of different orders Dastbasteh, Reza Padashnick, Farzad Crespo, Pedro M. Grassl, Markus Sharafi, Javad Information Theory 94B05, 94B15 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. |
| title | Equivalence of constacyclic codes with shift constants of different orders |
| topic | Information Theory 94B05, 94B15 |
| url | https://arxiv.org/abs/2403.04600 |