Hadamard full propelinear codes of type Q; rank and kernel

© 2017, Springer Science+Business Media, LLC. Hadamard full propelinear codes (HFP -codes) are introduced and their equivalence with Hadamard groups is proven; on the other hand, the equivalence of Hadamard groups, relative (4n, 2, 4n, 2n)-difference sets in a group, and cocyclic Hadamard matrices, is already known. We compute the available values for the rank and dimension of the kernel of HFP -codes of type Q and we show that the dimension of the kernel is always 1 or 2. We also show that when the dimension of the kernel is 2 then the dimension of the kernel of the transposed code is 1 (so, both codes are not equivalent). Finally, we give a construction method such that from an HFP -code of length 4n, dimension of the kernel k= 2 , and maximum rank r= 2 n, we obtain an HFP -code of double length 8n, dimension of the kernel k= 2 , and maximum rank r= 4 n.