On generalized Legendre pairs and multipliers of the corresponding supplementary difference sets

Abstract

The autocorrelation function of a sequence is a measure for how much the given sequence differs from its translates. Binary sequences with good periodic autocorrelation properties have important applications in various areas of engineering. They are also used in several topics of combinatorics. In particular, sometimes one needs two ±1 sequences for which the sum of their periodic autocorrelations, except for the O-th term, is a constant, say γ. If γ = -2 these two sequences are called generalized Legendre pairs. In this paper, we consider generalized Legendre pairs, which are presented in the form of the corresponding supplementary difference sets. We investigate multipliers of these supplementary difference sets, and we construct a series of such pairs. The construction is achieved through an algorithm which is also presented. Using some facts from group theory, this algorithm employs a fast searching method to find the multipliers and to construct the corresponding supplementary differences sets.