Lower Bound on the Minimum Distance of Single-Generator Quasi-Twisted Codes
We recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) codes....
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MDPI AG
2023-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/11/2539 |
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| author | Adel Alahmadi Patrick Solé Ramy Taki Eldin |
| author_facet | Adel Alahmadi Patrick Solé Ramy Taki Eldin |
| author_sort | Adel Alahmadi |
| collection | DOAJ |
| description | We recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) codes. Associating each QT code with a constacyclic code over an extension field, we obtain the first bound. This is the QT analogue to a result in the literature for quasi-cyclic codes. We point out some weaknesses in this bound and propose a novel bound that takes into account the Chinese remainder theorem approach to QT codes as well as the BCH bound of constacyclic codes. This proposed bound, in contrast to previous bounds in the literature, does not presuppose a specific form of code generator and does not require calculations in any extension field. We illustrate that our bound meets the one in the literature when the code generator adheres to the specific form assumed in that study. Various numerical examples enable us to compare and discuss these bounds. |
| first_indexed | 2024-03-11T03:01:31Z |
| format | Article |
| id | doaj.art-4895e374c8894539a131ca2c8cc48a45 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T03:01:31Z |
| publishDate | 2023-05-01 |
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| series | Mathematics |
| spelling | doaj.art-4895e374c8894539a131ca2c8cc48a452023-11-18T08:13:24ZengMDPI AGMathematics2227-73902023-05-011111253910.3390/math11112539Lower Bound on the Minimum Distance of Single-Generator Quasi-Twisted CodesAdel Alahmadi0Patrick Solé1Ramy Taki Eldin2Math Department, King Abdulaziz University, Jeddah 21589, Saudi ArabiaI2M, (Aix Marseille Univ., CNRS, Centrale Marseille), 13009 Marseilles, FranceFaculty of Engineering, Ain Shams University, Cairo 11517, EgyptWe recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) codes. Associating each QT code with a constacyclic code over an extension field, we obtain the first bound. This is the QT analogue to a result in the literature for quasi-cyclic codes. We point out some weaknesses in this bound and propose a novel bound that takes into account the Chinese remainder theorem approach to QT codes as well as the BCH bound of constacyclic codes. This proposed bound, in contrast to previous bounds in the literature, does not presuppose a specific form of code generator and does not require calculations in any extension field. We illustrate that our bound meets the one in the literature when the code generator adheres to the specific form assumed in that study. Various numerical examples enable us to compare and discuss these bounds.https://www.mdpi.com/2227-7390/11/11/2539quasi-twisted codesconstacyclic codesquasi-cyclic codesBCH bound |
| spellingShingle | Adel Alahmadi Patrick Solé Ramy Taki Eldin Lower Bound on the Minimum Distance of Single-Generator Quasi-Twisted Codes Mathematics quasi-twisted codes constacyclic codes quasi-cyclic codes BCH bound |
| title | Lower Bound on the Minimum Distance of Single-Generator Quasi-Twisted Codes |
| title_full | Lower Bound on the Minimum Distance of Single-Generator Quasi-Twisted Codes |
| title_fullStr | Lower Bound on the Minimum Distance of Single-Generator Quasi-Twisted Codes |
| title_full_unstemmed | Lower Bound on the Minimum Distance of Single-Generator Quasi-Twisted Codes |
| title_short | Lower Bound on the Minimum Distance of Single-Generator Quasi-Twisted Codes |
| title_sort | lower bound on the minimum distance of single generator quasi twisted codes |
| topic | quasi-twisted codes constacyclic codes quasi-cyclic codes BCH bound |
| url | https://www.mdpi.com/2227-7390/11/11/2539 |
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