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On the kernel of \(\mathbb {Z}_{2^s}\)-linear Hadamard codes. (English) Zbl 1429.94082

Barbero, Ángela I. (ed.) et al., Coding theory and applications. 5th international castle meeting, ICMCTA 2017, Vihula, Estonia, August 28–31, 2017. Proceedings. Cham: Springer. Lect. Notes Comput. Sci. 10495, 107-117 (2017).
Summary: The \(\mathbb {Z}_{2^s}\)-additive codes are subgroups of \(\mathbb {Z}^n_{2^s}\), and can be seen as a generalization of linear codes over \(\mathbb {Z}_2\) and \(\mathbb {Z}_4\). A \(\mathbb {Z}_{2^s}\)-linear Hadamard code is a binary Hadamard code which is the Gray map image of a \(\mathbb {Z}_{2^s}\)-additive code. For \(s=2\), the kernel of \(\mathbb {Z}_4\)-linear Hadamard codes can be used for their complete classification. In this paper, the kernel of \(\mathbb {Z}_{2^s}\)-linear Hadamard codes is given for \(s > 2\). However, unlike for \(s=2\), we show that the dimension of the kernel just allows a partial classification of these Hadamard codes.
For the entire collection see [Zbl 1370.94004].

MSC:

94B05 Linear codes (general theory)
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