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

The combinatorics of binary arrays

This paper gives an account of the combinatorics of binary arrays, mainly concerning their randomness properties. In many cases the problem reduces to the investigation on difference sets.postprin

Binary matrices of optimal autocorrelations as alignment marks

We define a new class of binary matrices by maximizing the peak-sidelobe distances in the aperiodic autocorrelations. These matrices can be used as robust position marks for in-plane spatial alignment. The optimal square matrices of dimensions up to 7 by 7 and optimal diagonally-symmetric matrices of 8 by 8 and 9 by 9 were found by exhaustive searches.Comment: 8 pages, 6 figures and 1 tabl

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.

Hadamard matrices modulo 5

In this paper we introduce modular symmetric designs and use them to study the existence of Hadamard matrices modulo 5. We prove that there exist 5-modular Hadamard matrices of order n if and only if n != 3, 7 (mod 10) or n != 6, 11. In particular, this solves the 5-modular version of the Hadamard conjecture.Comment: 7 pages, submitted to JC