Hadamard ideals and Hadamard matrices with two circulant cores

We apply Computational Algebra methods to the construction of Hadamard matrices with two circulant cores, given by Fletcher, Gysin and Seberry. We introduce the concept of Hadamard ideal for this construction to systematize the application of Computational Algebra methods. Our approach yields an exhaustive search construction of Hadamard matrices with two circulant cores for this construction for the ten first admissible values 3, 5, 7, 9, 11, 13, 15, 17, 19, 21. For each of these ten parameter values, the number of non-equivalent such Hadamard matrices is proportional to the square of the parameter.