Gray isometries for finite $p$-groups
We construct two classes of Gray maps, called type-I Gray map and type-II Gray map, for a finite $p$-group $G$. Type-I Gray maps are constructed based on the existence of a Gray map for a maximal subgroup $H$ of $G$. When $G$ is a semidirect product of two finite $p$-groups $H$ and $K$, both $H$ and...
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| Format: | Article |
| Language: | English |
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University of Isfahan
2013-03-01
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| Series: | Transactions on Combinatorics |
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| Online Access: | http://www.combinatorics.ir/?_action=showPDF&article=2762&_ob=c4b07819ea8798454cff55f3b453576c&fileName=full_text.pdf. |
| Summary: | We construct two classes of Gray maps, called type-I Gray map and type-II Gray map, for a finite $p$-group $G$. Type-I Gray maps are constructed based on the existence of a Gray map for a maximal subgroup $H$ of $G$. When $G$ is a semidirect product of two finite $p$-groups $H$ and $K$, both $H$ and $K$ admit Gray maps and the corresponding homomorphism $psi:Hlongrightarrow {rm Aut}(K)$ is compatible with the Gray map of $K$ in a sense which we will explain, we construct type-II Gray maps for $G$. Finally, we consider group codes over the dihedral group $D_8$ of order 8 given by the set of their generators, and derive a representation and an encoding procedure for such codes. |
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| ISSN: | 2251-8657 2251-8665 |