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Algebra and Discrete Mathematics, 2016, том 21, выпуск 1, страницы 59–68
(Mi adm554)
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Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)
RESEARCH ARTICLE
Construction of self-dual binary $[2^{2k},2^{2k-1},2^k]$-codes
Carolin Hannusch, Piroska Lakatos Institute of Mathematics, University of Debrecen, 4010 Debrecen, pf.12, Hungary
Аннотация:
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 [2]). 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.
Ключевые слова:
Reed–Muller code, Generalized Reed–Muller code, radical, self-dual code, group algebra, Jacobson radical.
Поступила в редакцию: 21.09.2015 Исправленный вариант: 16.12.2015
Образец цитирования:
Carolin Hannusch, Piroska Lakatos, “Construction of self-dual binary $[2^{2k},2^{2k-1},2^k]$-codes”, Algebra Discrete Math., 21:1 (2016), 59–68
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/adm554 https://www.mathnet.ru/rus/adm/v21/i1/p59
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