A Recursive Construction for Skew Hadamard Difference Sets

A major conjecture on the existence of abelian skew Hadamard difference sets is: if an abelian group $G$ contains a skew Hadamard difference set, then $G$ must be elementary abelian. This conjecture remains open in general. In this paper, we give a recursive construction for skew Hadamard difference sets in abelian (not necessarily elementary abelian) groups. The new construction can be considered as a result on the aforementioned conjecture: if there exists a counterexample to the conjecture, then there exist infinitely many counterexamples to it.

Quasi-Perfect Sequences and Hadamard Difference Sets

In this paper, we prove that a binary sequence is perfect (resp., quasi-perfect) if and only if its support set for any finite group (not necessarily cyclic) is a Hadamard difference set of type I (resp., type II); and we prove that the kernel of any nonzero linear functional (or the image of any linear transformation A with dim ( Ker A) = 1) on the linear space GF(2m) over the field GF(2) (excluding 0) is a cyclic Hadamard difference set of type II using Gaussian sums; and we prove that the multiplier group of the above difference set is equal to the Galois group Gal (GF(2m) / GF(2)); and we mention the relationship between the Hadamard transform and the irreducible complex characters.