A practical algorithm for completing half-Hadamard matrices using LLL

A (classical) partial Hadamard Matrix is an m× n matrix H with values in { - 1 , 1 } such that HH^T= nI_m. If m= n, we say that H is a (full) Hadamard Matrix. In this paper, we address the following Completion Problem: Can a given partial Hadamard matrix be completed to a full Hadamard matrix? And is this completion unique? As soon as m≥ n/ 2 , and under reasonable assumptions, we give a practical algorithm for finding the completion if it exists, or deciding that there is no completion with a great level of certainty. The algorithm works well in practice, at least for n< 150 , and depends on an unproven conjecture on the performance of the LLL algorithm. We apply a version of this algorithm to solve the Gram square-root problem: Solving XX^T= G for X over the integers. As an application, we show how to reconstruct a weighing matrix W(23, 16) from a 13 × 13 submatrix.