On the ground states of the Bernasconi model

The ground states of the Bernasconi model are binary +1/-1 sequences of length N with low autocorrelations. We introduce the notion of perfect sequences, binary sequences with one-valued off-peak correlations of minimum amount. If they exist, they are ground states. Using results from the mathematical theory of cyclic difference sets, we specify all values of N for which perfect sequences do exist and how to construct them. For other values of N, we investigate almost perfect sequences, i.e. sequences with two-valued off-peak correlations of minimum amount. Numerical and analytical results support the conjecture that almost perfect sequences do exist for all values of N, but that they are not always ground states. We present a construction for low-energy configurations that works if N is the product of two odd primes.Comment: 12 pages, LaTeX2e; extended content, added references; submitted to J.Phys.

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

An Efficient Joint Source-Channel Decoder with Dynamical Block Priors

An efficient joint source-channel (s/c) decoder based on the side information of the source and on the MN-Gallager algorithm over Galois fields is presented. The dynamical block priors (DBP) are derived either from a statistical mechanical approach via calculation of the entropy for the correlated sequences, or from the Markovian transition matrix. The Markovian joint s/c decoder has many advantages over the statistical mechanical approach. In particular, there is no need for the construction and the diagonalization of a qXq matrix and for a solution to saddle point equations in q dimensions. Using parametric estimation, an efficient joint s/c decoder with the lack of side information is discussed. Besides the variant joint s/c decoders presented, we also show that the available sets of autocorrelations consist of a convex volume, and its structure can be found using the Simplex algorithm.Comment: 13 pages, to appear in "Progress in Theoretical Physics Supplement", May 200

Generation of unpredictable time series by a Neural Network

A perceptron that learns the opposite of its own output is used to generate a time series. We analyse properties of the weight vector and the generated sequence, like the cycle length and the probability distribution of generated sequences. A remarkable suppression of the autocorrelation function is explained, and connections to the Bernasconi model are discussed. If a continuous transfer function is used, the system displays chaotic and intermittent behaviour, with the product of the learning rate and amplification as a control parameter.Comment: 11 pages, 14 figures; slightly expanded and clarified, mistakes corrected; accepted for publication in PR

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

Low autocorrelated multi-phase sequences

The interplay between the ground state energy of the generalized Bernasconi model to multi-phase, and the minimal value of the maximal autocorrelation function, $C_{max}=\max_K{|C_K|}$, $K=1,..N-1$, is examined analytically and the main results are: (a) The minimal value of $\min_N{C_{max}}$ is $0.435\sqrt{N}$ significantly smaller than the typical value for random sequences $O(\sqrt{\log{N}}\sqrt{N})$. (b) $\min_N{C_{max}}$ over all sequences of length N is obtained in an energy which is about 30% above the ground-state energy of the generalized Bernasconi model, independent of the number of phases m. (c) The maximal merit factor $F_{max}$ grows linearly with m. (d) For a given N, $\min_N{C_{max}}\sim\sqrt{N/m}$ indicating that for m=N, $\min_N{C_{max}}=1$, i.e. a Barker code exits. The analytical results are confirmed by simulations.Comment: 4 pages, 4 figure

Binary time series generated by chaotic logistic maps

This paper examines stochastic pairwise dependence structures in binary time series obtained from discretised versions of standard chaotic logistic maps. It is motivated by applications in communications modelling which make use of so-called chaotic binary sequences. The strength of non-linear stochastic dependence of the binary sequences is explored. In contrast to the original chaotic sequence, the binary version is non-chaotic with non-Markovian non-linear dependence, except in a special case. Marginal and joint probability distributions, and autocorrelation functions are elicited. Multivariate binary and more discretised time series from a single realisation of the logistic map are developed from the binary paradigm. Proposals for extension of the methodology to other cases of the general logistic map are developed. Finally, a brief illustration of the place of chaos-based binary processes in chaos communications is given.Binary sequence; chaos; chaos communications; dependence; discretisation; invariant distribution; logistic map; randomness