Application of the discrete Fourier transform to the search for generalised Legendre pairs and Hadamard matrices

We introduce Legendre sequences and generalised Legendre pairs (G Lpairs). We show how to construct an Hadamard matrix of order 2£ + 2 from a GL-pair of length f. We review the known constructions for GLpairs and use the discrete Fourier transform (DFT) and power spectral density (PSD) to enable an exhaustive search for GL-pairs for lengths f ::::; 47 and partial searches for other f, 1 Definitions and Notation Let U be a sequence of e real numbers Uo, Ul, .'" U£-l' The periodic autocorrelation junction Pu(j) of such a sequence is defined by: £-1 Pu(j) = L: UiUi+j mod £, j = 0,1, "" f 1. i=O * Research supported by ARC Large Grants A9803826 and A49703117. Australasian Journal of Combinatorics 23(2001), pp.75-86 Two sequences U and V of identical length £ are said to be compatible if the sum of their periodic auto correlations is a constant, say a, except for the O-th term. That is, Pu(j) + Pv(j) = a, j =I O. (1) (Such pairs are said to have constant periodic autocorrelation even though it is the sum of the auto correlations that is a constant.) If U and V are both ±1 sequences, compatible and a = -2, then they are called a generalised Legendre pair (or GLpair) of length £. We will denote a GL-pair of length £ by GL(£). In Sections 3-5, we restrict our attention to G L-pairs. An Hadamard matrix of order n is an n x n matrix H which has ±l-entries and all its rows and columns are orthogonal. In other words HHT = nIn where In is the identity matrix of order n. For the definition of supplementary difference sets the reader is referred to [WSW72]. We note that two compatible sequences may contain elements from any alphabet. If the elements of two compatible sequences are 0,1 then they are described as 2 {f; kl' k2 ; ),} supplementary difference sets (SDS). In this paper we are interested in the particular case of 2 {f; £~1, £~1; £~1} SDS since these give, when the zeros are replaced by -1, compatible ±1 sequences which are a GL(£)-pair, and may be used as below to construct Hadamard matrices of order 2£ + 2. The Legendre or Jacobi symbol is written (aln) if n is prime or composite, respectively. When referring to the elements of a-I, 0,1 sequence we often write '-' instead of -1 and '+' instead of 1. The discrete Fourier transform (DFT) of a sequence U is given by £-1 DFTu(k) = fJk = L UiWik , k = 0,1, ... , £ 1 i=O where w is the primitive £-th root of unity e 2ii. If we take the squared magnitude of each term in the DFT of U, the resulting sequence is called the power spectral density (PSD) of U. The k-th terms in the PSDs of U and V are denoted by IfJkl2 and IVkI2. Example 1 The PSD of the sequence 1 2 2 -2 0 0 0 is 9.000 19.988 13.220 7.792 7.792 13.220 19.988. If a sequence U is transformed by the operation of cyclically taking every d-th element, where gcd(d, £) = 1, the sequence U is said to be decimated by d. That is, if V = U decimated by d, then Vi = Udi mod £.