The author provides a short introduction to the considered problem, then a preliminary section gives the basic properties of the complex quadratic residue codes and fixes notation. A theorem for computing determinants of Legendre symbol matrices is presented along with its proof. This method can also be used to compute other determinants, including the determinant of a \(D_p\) matrix, closely related to \(N_p\). Existing work in the field is the topic for the closing section with an original description of the proof.