In this paper, we illustrate the relationship between profiles of Hadamard matrices and weight distributions of codes, give a new and efficient method to determine the minimum weight of doubly even self-dual codes constructed by using Hadamard matrices of order with , and present a new proof that the codes have for all types of Hadamard matrices of order with . Finally, we discuss doubly even self-dual codes with or constructed by using all currently known Hadamard matrices of order .