Quasi-perfect sequences and Hadamard difference sets

Abstract

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(2(m)) over the field GF(2(m)) (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(2(m))/GF(2)); and we mention the relationship between the Hadamard transform and the irreducible complex characters.