Affine difference sets of even order

AbstractGeneralizing a result of Ko and Ray-Chaudhuri (Discrete Math. 39 (1982), 37–58), we show the following: Assume the existence of an affine difference set in G relative to N of even order n≠2. If G is of the form G = N ⊕ H, where N is abelian, then n is actually a multiple of 4, say n = 4k, and there exists a (4k − 1, 2k − 1, k − 1)-Hadamard difference set in N. More detailed considerations lead to variations of this result (under appropriate assumptions) which yield even stronger non-existence theorems. In particular, we show the non-existence of abelian affine difference sets of order n ≡ 4 mod 8 (with the exception n = 4) and of nilpotent affine difference sets of order n ≡ 2 mod 4 (n ≠ 2). The latter result is the first general non-existence theorem in the non-abelian case