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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 \(d\) of doubly even self-dual \([2n,n,d]\) codes constructed by using Hadamard matrices of order \(n = 8t + 4\) with \(t \geq 1\), and present a new proof that the \([2n,n,d]\) codes have \(d \geq 8\) for all types of Hadamard matrices of order \(n = 8t + 4\) with \(t \geq 1\). Finally, we discuss doubly even self-dual \([72,36,d]\) codes with \(d = 8\) or \(d = 12\) constructed by using all currently known Hadamard matrices of order \(n = 36\).