Families of Hadamard $BBZ_{2}BBZ_{4}Q_{8}$ -Codes

A Z₂Z₄Q₈-code is the binary image, after a Gray map, of a subgroup of Z₂^k₁ × Z₄^k2 × Q₈^k3, where Q₈ is the quaternion group on eight elements. Such Z₂Z₄Q₈-codes are translation invariant propelinear codes as are the well known Z₄-linear or Z₂Z₄-linear codes. In this paper, we show that there exist “pure” Z₂Z₄Q₈-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the Z₂Z₄Q₈-codes, and we give upper and lower bounds for these parameters. We give tools to construct a new class of Hadamard codes formed by several families of Z₂Z₄Q₈-codes; we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are Hadamard.