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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Lugansk National Taras Shevchenko University
2016
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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 |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2016-05-11T05:58:11Z |
| collection |
OJS |
| language |
English |
| topic |
Reed--Muller code Generalized Reed--Muller code radical self-dual code; group algebra; Jacobson radical 94B05 11T71 20C05 |
| 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 |
| topic_facet |
Reed--Muller code Generalized Reed--Muller code radical self-dual code; group algebra; Jacobson radical 94B05 11T71 20C05 |
| format |
Article |
| author |
Hannusch, Carolin Lakatos, Piroska |
| author_facet |
Hannusch, Carolin Lakatos, Piroska |
| author_sort |
Hannusch, Carolin |
| title |
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_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_sort |
construction of self-dual binary \([2^{2k},2^{2k-1},2^k]\)-codes |
| 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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2016 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/25 |
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AT hannuschcarolin constructionofselfdualbinary22k22k12kcodes AT lakatospiroska constructionofselfdualbinary22k22k12kcodes |
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2025-12-02T15:39:55Z |
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2025-12-02T15:39:55Z |
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