# Gray isometries for finite $p$-groups

Document Type: Research Paper

Author

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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### References

A. A. Nechaev (1991). Kerdock code in a cyclic form. Discrete Math. Appl.. 1, 365-384
A. R. Hammons, P. V. Kummar, A. R. Calderbank, N. J. A. Sloane and P. Sole (1994). The ${mathbb Z}_4$ linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory. 40, 301-319
A. R. Calderbank and G. McGuire (1997). Construction of a $(64,2^{37},12)$ code via Galois rings. Des. Codes Cryptogr.. 10, 157-165
M. Greferath and S. E. Schmidt (1999). Gray isometries for finite chain rings and a nonlinear ternary $(36,3^{12},15)$-code. IEEE Trans. Inform. Theory. 45, 2522-2524
I. M. Duursma, M. Greferath, S. Litsyn, and S. E. Schmidt (2001). A ${mathbb Z}_8$-linear lift of the binary Golay code and a non-linear binary $(96, 2^{37}, 24)$-code. IEEE Trans. Inform. Theory. 47, 1596-1598
M. Kiermaier and J. Zwanzger (2011). A ${mathbb Z}_4$-linear code of high minimum Lee distance derived from a hyperoval. Adv. Math. Commun.. 5, 275-286
I. Constantinescu and W. Heise (1997). A metric for codes over residue class rings. Problems Inform. Transmission. 33, 208-213
H. Tapia-Recillas and G. Vega (2003). Some constacyclic codes over ${mathbb Z}_{2^k}$ and binary quasi-cyclic codes. Discrete Appl. Math.. 128, 305-316
S. Ling and T. Blackford (2002). ${mathbb Z}_{p^{k+1}}$-linear codes. IEEE Trans. Inform. Theory. 48, 2592-2605
J. F. Qian, L. N. Zhang and S. X. Zhu (2006). $(1+u)$ Constacyclic and cyclic codes over ${mathbb F}_2+u{mathbb F}_2$. Appl. Math. Lett.. 19, 820-823
J. F. Qian, L. N. Zhang and S. X. Zhu (2006). Constacyclic and cyclic codes over ${mathbb F}_2+u{mathbb F}_2+u^2{mathbb F}_2$. IEICE Trans. Fundamentals. E89-A, 1863-1865
R. Sobhani and M. Esmaeili (2010). Some Constacyclic and cyclic codes over ${mathbb F}_q[u]/left$. IEICE Trans. Fundamentals. E93-A, 808-813
G. D. Forney (1992). On the Hamming distance properties of group codes. IEEE Trans. Inform. Theory. 38, 1797-1801