A Fourier-analytic Approach to Counting Partial Hadamard Matrices

Abstract

In this paper, we study a family of lattice walks which are related to the Hadamard conjecture. There is a bijection between paths of these walks which originate and terminate at the origin and equivalence classes of partial Hadamard matrices. Therefore, the existence of partial Hadamard matrices can be proved by showing that there is positive probability of a random walk returning to the origin after a specified number of steps. Moreover, the number of these designs can be approximated by estimating the return probabilities. We use the inversion formula for the Fourier transform of the random walk to provide such estimates. We also include here an upper bound, derived by elementary methods, on the number of partial Hadamard.Comment: \keywords{partial Hadamard matrices, random walks} Submitted to Cryptography and Communications: Discrete Structures, Boolean Functions and Sequences (Special Issue on Design Theory) March 18th 2010