When the necessary conditions are not sufficient: sequences with zero autocorrelation function

Recently K. T. Arasu (personal communication) and Yoseph Strassler, in his PhD thesis, The Classification of Circulant Weighing Matrices of Weight 9, Bar-Ilan University, Ramat-Gan, 1997, have intensively studied circulant weighing matrices, or single sequences, with weight 9. They show many cases are non-existent. Here we give details of a search for two sequences with zero periodic autocorrelation and types (1,9), (1,16) and (4,9). We find some new cases but also many cases where the known necessary conditions are not sufficient. We instance a number of occasions when the known necessary conditions are not sufficient for the existence of weighing matrices and orthogonal de-signs constructed using sequences with zero autocorrelation function leading to intriguing new questions

Do the Barker Codes End?

A Barker code is a binary code with k^th autocorrelation <= 1 for all nonzero k. At the workshop, the Barker code group split into four non-disjoint subgroups: - An "algebra group", who explored symmetries of the search space that preserve the autocorrelations' magnitude. - A "computing group", who explored methods for quickly finding binary codes with very good autocorrelation properties. - A "statistics group", who explored ways to quantify what has been empirically observed about autocorrelation in the search space S_2^N. - A "continuous group", who explored a non-discrete analogue of the problem of finding sequences with good autocorrelations

Optimization Methods for Designing Sequences with Low Autocorrelation Sidelobes

Unimodular sequences with low autocorrelations are desired in many applications, especially in the area of radar and code-division multiple access (CDMA). In this paper, we propose a new algorithm to design unimodular sequences with low integrated sidelobe level (ISL), which is a widely used measure of the goodness of a sequence's correlation property. The algorithm falls into the general framework of majorization-minimization (MM) algorithms and thus shares the monotonic property of such algorithms. In addition, the algorithm can be implemented via fast Fourier transform (FFT) operations and thus is computationally efficient. Furthermore, after some modifications the algorithm can be adapted to incorporate spectral constraints, which makes the design more flexible. Numerical experiments show that the proposed algorithms outperform existing algorithms in terms of both the quality of designed sequences and the computational complexity

Autocorrelations of Binary Sequences and Run Structure

We analyze the connection between the autocorrelation of a binary sequence and its run structure given by the run length encoding. We show that both the periodic and the aperiodic autocorrelation of a binary sequence can be formulated in terms of the run structure. The run structure is given by the consecutive runs of the sequence. Let C=(C(0), C(1),...,C(n)) denote the autocorrelation vector of a binary sequence. We prove that the kth component of the second order difference operator of C can be directly calculated by using the consecutive runs of total length k. In particular this shows that the kth autocorrelation is already determined by all consecutive runs of total length L<k. In the aperiodic case we show how the run vector R can be efficiently calculated and give a characterization of skew-symmetric sequences in terms of their run length encoding.Comment: [v3]: minor revisions, accepted for publication in IEEE Trans. Inf. Theory, 17 page