Rank and kernel of binary Hadamard codes

In this paper, the rank and the dimension of the kernel for (binary) Hadamard codes of length a power of two are studied. In general, it is well-known that the rank of a Hadamard code of length n=2^t is a value in {t+1,...,n/2}. In the present paper, the range of possible values for the dimension of the kernel is computed and a construction of Hadamard codes of length n=2^t for each one of these values is given. Lower and upper bounds for the rank and dimension of the kernel of a Hadamard code of length n=2^t are also established. Finally, we construct Hadamard codes for all possible ranks and dimension of kernels between these bounds.