A new family of Hadamard matrices of order $4(2q^2+1)$

Let $q$ be a prime power of the form $q=12c^2+4c+3$ with $c$ an arbitrary integer. In this paper we construct a difference family with parameters $(2q^2;q^2,q^2,q^2,q^2-1;2q^2-2)$ in ${mathbb Z}_2times ({mathbb F}_{q^2},+)$. As a consequence, by applying the Wallis-Whiteman array, we obtain Hadamard matrices of order $4(2q^2+1)$ for the aforementioned $q$s.