$\mathbb {Z}_{2}\mathbb {Z}_{2}[u]$ –Cyclic and Constacyclic Codes
| Autor: | Irfan Siap, Ismail Aydogdu, Taher Abualrub |
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| Rok vydání: | 2017 |
| Předmět: |
Discrete mathematics
Ring (mathematics) 020206 networking & telecommunications 0102 computer and information sciences 02 engineering and technology Library and Information Sciences 01 natural sciences Computer Science Applications Combinatorics 010201 computation theory & mathematics 0202 electrical engineering electronic engineering information engineering Dual polyhedron Information Systems Mathematics |
| Zdroj: | IEEE Transactions on Information Theory. 63:4883-4893 |
| ISSN: | 1557-9654 0018-9448 |
| Popis: | Following the very recent studies on $\mathbb {Z}_{2}\mathbb {Z}_{4}$ -additive codes, $\mathbb {Z}_{2}\mathbb {Z}_{2}[u]$ -linear codes have been introduced by Aydogdu et al. In this paper, we introduce and study the algebraic structure of cyclic, constacyclic codes and their duals over the $R$ -module $\mathbb {Z}_{2}^\alpha R^\beta $ where $R=\mathbb {Z}_{2}+u\mathbb {Z}_{2}=\left \{{0,1,u,u+1}\right \}$ is the ring with four elements and $u^{2}=0$ . We determine the generating independent sets and the types and sizes of both such codes and their duals. Finally, we present a bound and an optimal family of codes attaining this bound and also give some illustrative examples of binary codes that have good parameters which are obtained from the cyclic codes in $\mathbb {Z}_{2}^\alpha R^\beta $ . |
| Databáze: | OpenAIRE |
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