On $Z_{2^k}$-Dual Binary Codes

Details On $Z_{2^k}$-Dual Binary Codes On $Z_{2^k}$-Dual Binary Codes

2005-09-14

arxiv.org/abs/math/0509325v3

IEEE Trans. Inf. Theory 53(4) 2007, 1532-1537

Mathematics

Combinatorics

English: 10pp, Russian: 14pp; V.1 title: Z_{2^k}-duality, Z_{2^k}-linear Hadamard codes, and co-Z_{2^k}-linear 1-perfect codes

Scientific paper

10.1109/TIT.2007.892787

A new generalization of the Gray map is introduced. The new generalization $\Phi: Z_{2^k}^n \to Z_{2}^{2^{k-1}n}$ is connected with the known generalized Gray map $\phi$ in the following way: if we take two dual linear $Z_{2^k}$-codes and construct binary codes from them using the generalizations $\phi$ and $\Phi$ of the Gray map, then the weight enumerators of the binary codes obtained will satisfy the MacWilliams identity. The classes of $Z_{2^k}$-linear Hadamard codes and co-$Z_{2^k}$-linear extended 1-perfect codes are described, where co-$Z_{2^k}$-linearity means that the code can be obtained from a linear $Z_{2^k}$-code with the help of the new generalized Gray map. Keywords: Gray map, Hadamard codes, MacWilliams identity, perfect codes, $Z_{2^k}$-linearity

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