Skew Hadamard difference families and skew Hadamard matrices

In this paper, we generalize classical constructions of skew Hadamard difference families with two or four blocks in the additive groups of finite fields given by Szekeres (1969, 1971), Whiteman (1971) and Wallis-Whiteman (1972). In particular, we show that there exists a skew Hadamard difference family with $2^{u-1}$ blocks in the additive group of the finite field of order $q^e$ for any prime power $q\equiv 2^u+1\,({\mathrm{mod\, \, }2^{u+1}})$ with $u\ge 2$ and any positive integer $e$. In the aforementioned work of Szekeres, Whiteman, and Wallis-Whiteman, the constructions of skew Hadamard difference families with $2^{u-1}$ ($u=2$ or $3$) blocks in $({\mathbb F}_{q^e},+)$ depend on the exponent $e$, with $e\equiv 1,2,$ or $3\,({\mathrm{mod\, \, }4})$ when $u=2$, and $e\equiv 1\,({\mathrm{mod\, \, }2})$ when $u=3$, respectively. Our more general construction, in particular, removes the dependence on $e$. As a consequence, we obtain new infinite families of skew Hadamard matrices.