Gray isometries for finite $p$-groups

Document Type: Research Paper

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Abstract

‎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 $\psi:H\longrightarrow {\rm‎ ‎Aut}(K)$ 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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