Kronecker sums to construct Hadamard full propelinear codes of type CnQ8

Hadamard matrices with a subjacent algebraic structure have been deeply studied as well as the links with other topics in algebraic combinatorics [1]. An important and pioneering paper about this subject is [5], where it is introduced the concept of Hadamard group. In addition, we find beautiful equivalences between Hadamard groups, 2-cocyclic matrices and relative difference sets [4], [7]. From the side of coding theory, it is desirable that the algebraic structures we are dealing with preserves the Hamming distance. This is the case of the propelinear codes and specially those which are translation invariant which has been characterized as the image, by a suitable Gray map, of a subgroup of a direct product of Z2, Z4 and Q8 (see [8] and references there). As for the 2-cocyclic matrices and relative difference sets it was shown in [10] that the concept of Hadamard group is equivalent to Hadamard full propelinear codes (HFP for short). This new equivalence provides a good place to study the rank and the dimension of the kernel of the Hadamard codes we construct. These are important steps trying to solve several conjectures involving Hadamard matrices. In [6] it was introduced a special Hadamard group, called type Q and it was conjectured that Hadamard matrices of this type exists for all possible lengths. In this paper we are studying Hadamard codes of type CnQ8, which are full propelinear and the subjacent group structure is isomorphic to a direct sum of the cyclic group Cn and the quaternion group Q8. The main results we present are about the links with the Hadamard codes of type Q and the construction of Kronecker sums allowing to duplicate or quadruplicate the length of the code. With the current results we conjecture that it is not possible to go deeper with the Kronecker construction than duplicate or quadruplicate the initial HFP-code, otherwise we contradicts the Ryse conjecture [11] about circulant Hadamard matrices.