On a new class of productive regular Hadamard matrices

We show that for each integer n for which there is a Hadamard matrix of order 4n and 8n²-1 is a prime number, there is a productive regular Hadamard matrix of order 16n²(8n²-1)². As a corollary, by applying a recent result of Ionin, we get many parametrically new classes of symmetric designs whenever either of 4n(8n²-1)-1 or 4n(8n²-1)+1 is a prime power.