Asymptotic enumeration by degree sequence of graphs of high degree
European Journal of Combinatorics
Signed groups, sequences, and the asymptotic existence of Hadamard matrices
Journal of Combinatorial Theory Series A
A comment on the Hadamard conjecture
Journal of Combinatorial Theory Series A
On RSA moduli with almost half of the bits prescribed
Discrete Applied Mathematics
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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 matrices.