Constructions of skew Hadamard difference families

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 2u−1 blocks in the additive group of the finite field of order qe for any prime power q≡2u+1(mod2u+1) with u⩾2 and any positive integer e. In the aforementioned papers of Szekeres, Whiteman, and Wallis-Whiteman, the constructions of skew Hadamard difference families with 2u−1 (u=2 or 3) blocks in (Fqe,+) work only for restricted e;namely e≡1,2, or 3(mod4) when u=2, and e≡1(mod2) when u=3, respectively. Our more general construction, in particular, removes the restrictions on e. As a consequence, we obtain new infinite series of skew Hadamard difference families with two or four blocks, and hence skew Hadamard matrices.