Tightening Turyn’s bound for Hadamard difference sets

摘要

This work examines the existence of (4q 2,2q 2?q,q 2?q) difference sets, for q=p f , where p is a prime and f is a positive integer. Suppose that G is a group of order 4q 2 which has a normal subgroup K of order q such that G/K ? C q ×C 2×C 2, where C q ,C 2 are the cyclic groups of order q and 2 respectively. Under the assumption that p is greater than or equal to 5, this work shows that G does not admit (4q 2,2q 2?q,q 2?q) difference sets.