Construction of self-dual binary \([2^{2k},2^{2k-1},2^k]\)-codes
The binary Reed-Muller code \({\rm RM}(m-k,m)\) corresponds to the \(k\)-th power of the radical of \(GF(2)[G],\) where \(G\) is an elementary abelian group of order \(2^m \) (see~\cite{B}). Self-dual RM-codes (i.e. some powers of the radical of the previously mentioned group algebra) exist only for...
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| Дата: | 2016 |
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Lugansk National Taras Shevchenko University
2016
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543034153369600 |
|---|---|
| author | Hannusch, Carolin Lakatos, Piroska |
| author_facet | Hannusch, Carolin Lakatos, Piroska |
| author_sort | Hannusch, Carolin |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2016-05-11T05:58:11Z |
| description | The binary Reed-Muller code \({\rm RM}(m-k,m)\) corresponds to the \(k\)-th power of the radical of \(GF(2)[G],\) where \(G\) is an elementary abelian group of order \(2^m \) (see~\cite{B}). Self-dual RM-codes (i.e. some powers of the radical of the previously mentioned group algebra) exist only for odd \(m\).The group algebra approach enables us to find a self-dual code for even \(m=2k \) in the radical of the previously mentioned group algebra with similarly good parameters as the self-dual RM codes.In the group algebra\[ GF(2)[G]\cong GF(2)[x_1,x_2,\dots, x_m]/(x_1^2-1,x_2^2-1, \dots x_m^2-1)\]we construct self-dual binary \(C=[2^{2k},2^{2k-1},2^k]\) codes with property\[{\rm RM}(k-1,2k) \subset C \subset {\rm RM}(k,2k)\] for an arbitrary integer \(k\).In some cases these codes can be obtained as the direct product of two copies of \({\rm RM}(k-1,k)\)-codes. For \(k\geq 2\) the codes constructed are doubly even and for \(k=2\) we get two non-isomorphic \([16,8,4]\)-codes. If \(k>2\) we have some self-dual codes with good parameters which have not been described yet. |
| first_indexed | 2025-12-02T15:39:55Z |
| format | Article |
| id | admjournalluguniveduua-article-25 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:39:55Z |
| publishDate | 2016 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-252016-05-11T05:58:11Z Construction of self-dual binary \([2^{2k},2^{2k-1},2^k]\)-codes Hannusch, Carolin Lakatos, Piroska Reed--Muller code, Generalized Reed--Muller code, radical, self-dual code; group algebra; Jacobson radical 94B05; 11T71; 20C05 The binary Reed-Muller code \({\rm RM}(m-k,m)\) corresponds to the \(k\)-th power of the radical of \(GF(2)[G],\) where \(G\) is an elementary abelian group of order \(2^m \) (see~\cite{B}). Self-dual RM-codes (i.e. some powers of the radical of the previously mentioned group algebra) exist only for odd \(m\).The group algebra approach enables us to find a self-dual code for even \(m=2k \) in the radical of the previously mentioned group algebra with similarly good parameters as the self-dual RM codes.In the group algebra\[ GF(2)[G]\cong GF(2)[x_1,x_2,\dots, x_m]/(x_1^2-1,x_2^2-1, \dots x_m^2-1)\]we construct self-dual binary \(C=[2^{2k},2^{2k-1},2^k]\) codes with property\[{\rm RM}(k-1,2k) \subset C \subset {\rm RM}(k,2k)\] for an arbitrary integer \(k\).In some cases these codes can be obtained as the direct product of two copies of \({\rm RM}(k-1,k)\)-codes. For \(k\geq 2\) the codes constructed are doubly even and for \(k=2\) we get two non-isomorphic \([16,8,4]\)-codes. If \(k>2\) we have some self-dual codes with good parameters which have not been described yet. Lugansk National Taras Shevchenko University 2016-05-10 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/25 Algebra and Discrete Mathematics; Vol 21, No 1 (2016) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/25/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/25/2 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/25/53 Copyright (c) 2016 Algebra and Discrete Mathematics |
| spellingShingle | Reed--Muller code Generalized Reed--Muller code radical self-dual code; group algebra; Jacobson radical 94B05 11T71 20C05 Hannusch, Carolin Lakatos, Piroska Construction of self-dual binary \([2^{2k},2^{2k-1},2^k]\)-codes |
| title | Construction of self-dual binary \([2^{2k},2^{2k-1},2^k]\)-codes |
| title_full | Construction of self-dual binary \([2^{2k},2^{2k-1},2^k]\)-codes |
| title_fullStr | Construction of self-dual binary \([2^{2k},2^{2k-1},2^k]\)-codes |
| title_full_unstemmed | Construction of self-dual binary \([2^{2k},2^{2k-1},2^k]\)-codes |
| title_short | Construction of self-dual binary \([2^{2k},2^{2k-1},2^k]\)-codes |
| title_sort | construction of self-dual binary \([2^{2k},2^{2k-1},2^k]\)-codes |
| topic | Reed--Muller code Generalized Reed--Muller code radical self-dual code; group algebra; Jacobson radical 94B05 11T71 20C05 |
| topic_facet | Reed--Muller code Generalized Reed--Muller code radical self-dual code; group algebra; Jacobson radical 94B05 11T71 20C05 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/25 |
| work_keys_str_mv | AT hannuschcarolin constructionofselfdualbinary22k22k12kcodes AT lakatospiroska constructionofselfdualbinary22k22k12kcodes |