  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.
