Families of Hadamard $\BBZ_{2}\BBZ_{4}Q_{8}$ -Codes
Ángel del Rı́o, Josep Rifà · IEEE Transactions on Information Theory · 2013
A$\BBZ_{2}\BBZ_{4}Q_{8}$-code is the binary image, after a Gray map, of a subgroup of$\BBZ_{2}^{k_{1}}\times\BBZ_{4}^{k_{2}}\times Q_{8}^{k_{3}}$, where$Q_{8}$is the quaternion group on eight elements. Such$\BBZ_{2}\BBZ_{4}Q_{8}$-codes are translation invariant propelinear codes as are the well known$\BBZ_{4}$-linear or$\BBZ_{2}\BBZ_{4}$-linear codes. In this paper, we show that there exist “pure”$\BBZ_{2}\BBZ_{4}Q_{8}$-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$\BBZ_{2}\BBZ_{4}Q_{8}$-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$\BBZ_{2}\BBZ_{4}Q_{8}$-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.