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, to systematize the application of computational algebra methods for this construction. We use the Hadamard ideal formalism to perform exhaustive search constructions of Hadamard matrices with two circulant cores for the twelve orders 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52. The total number of such Hadamard matrices is proportional to the square of the parameter. We use the Hadamard ideal formalism to compute the proportionality constants for the twelve orders listed above. Finally, we use the Hadamard ideal formalism to improve the lower bounds for the number of inequivalent Hadamard matrices for the seven orders 44, 48, 52, 56, 60, 64, 68.