Rank weight hierarchy of some classes of polynomial codes
We study the rank weight hierarchy, thus in particular the minimum rank distance, of polynomial codes over the finite field $\FF_{q^m}$, $q$ a prime power, $m \geq 2$. We assume that polynomials involved are squarefree. We establish the rank weight hierarchy of $[n,n-1]$ constacyclic codes. We chara...
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| Format: | Journal Article |
| Language: | English |
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2023
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| Online Access: | https://hdl.handle.net/10356/163959 |
| _version_ | 1824454889038675968 |
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| author | Ducoat, Jérôme Oggier, Frédérique |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Ducoat, Jérôme Oggier, Frédérique |
| author_sort | Ducoat, Jérôme |
| collection | NTU |
| description | We study the rank weight hierarchy, thus in particular the minimum rank distance, of polynomial codes over the finite field $\FF_{q^m}$, $q$ a prime power, $m \geq 2$. We assume that polynomials involved are squarefree. We establish the rank weight hierarchy of $[n,n-1]$ constacyclic codes. We characterize polynomial codes of $r$th rank weight $r$, and in particular of irst rank or minimum rank distance 1. Finally, we provide a refinement of the Singleton bound, from which we show that cyclic codes cannot be MRD (maximum rank distance) codes, but constacyclic codes can be. |
| first_indexed | 2025-02-19T03:29:29Z |
| format | Journal Article |
| id | ntu-10356/163959 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2025-02-19T03:29:29Z |
| publishDate | 2023 |
| record_format | dspace |
| spelling | ntu-10356/1639592023-02-28T20:09:24Z Rank weight hierarchy of some classes of polynomial codes Ducoat, Jérôme Oggier, Frédérique School of Physical and Mathematical Sciences Science::Mathematics Coding Theory Rank Metric We study the rank weight hierarchy, thus in particular the minimum rank distance, of polynomial codes over the finite field $\FF_{q^m}$, $q$ a prime power, $m \geq 2$. We assume that polynomials involved are squarefree. We establish the rank weight hierarchy of $[n,n-1]$ constacyclic codes. We characterize polynomial codes of $r$th rank weight $r$, and in particular of irst rank or minimum rank distance 1. Finally, we provide a refinement of the Singleton bound, from which we show that cyclic codes cannot be MRD (maximum rank distance) codes, but constacyclic codes can be. National Research Foundation (NRF) Submitted/Accepted version The early stage of this research by J. Ducoat and F. Oggier was supported by the Singapore National Research Foundation under Research Grant NRF-RF2009-07. 2023-01-06T00:31:53Z 2023-01-06T00:31:53Z 2022 Journal Article Ducoat, J. & Oggier, F. (2022). Rank weight hierarchy of some classes of polynomial codes. Designs, Codes and Cryptography. https://dx.doi.org/10.1007/s10623-022-01181-6 0925-1022 https://hdl.handle.net/10356/163959 10.1007/s10623-022-01181-6 en NRF-RF2009-07 Designs, Codes and Cryptography © 2022 The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature. All rights reserved. This version of the article has been accepted for publication, after peer review and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s10623-022-01181-6. application/pdf |
| spellingShingle | Science::Mathematics Coding Theory Rank Metric Ducoat, Jérôme Oggier, Frédérique Rank weight hierarchy of some classes of polynomial codes |
| title | Rank weight hierarchy of some classes of polynomial codes |
| title_full | Rank weight hierarchy of some classes of polynomial codes |
| title_fullStr | Rank weight hierarchy of some classes of polynomial codes |
| title_full_unstemmed | Rank weight hierarchy of some classes of polynomial codes |
| title_short | Rank weight hierarchy of some classes of polynomial codes |
| title_sort | rank weight hierarchy of some classes of polynomial codes |
| topic | Science::Mathematics Coding Theory Rank Metric |
| url | https://hdl.handle.net/10356/163959 |
| work_keys_str_mv | AT ducoatjerome rankweighthierarchyofsomeclassesofpolynomialcodes AT oggierfrederique rankweighthierarchyofsomeclassesofpolynomialcodes |