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This work deals with Hadamard Z2Z4Q8-codes, which are binary codes after aGray map from a subgroup of the direct product of Z2, Z4 and Q8 groups, whereQ8 is the quaternionic group. In a previous work, these kind of codes wereclassified in five shapes. In this paper we analyze the allowable range ofvalues for the rank and dimension of the kernel, which depends on theparticular shape of the code. We show that all these codes can be representedin a standard form, from a set of generators, which help to a wellunderstanding of the characteristics of each shape. The main results are thecharacterization of Hadamard Z2Z4Q8-codes as a quotient of a semidirect productof Z2Z4-linear codes and, on the other hand, the construction of HadamardZ2Z4Q8-codes code with any given pair of allowable parameters for the rank anddimension of the kernel.

Hadamard Z2Z4Q8-codes. Constructions based on the rank and dimension of the kernel