TY - JOUR

T1 - A practical algorithm for completing half-Hadamard matrices using LLL

AU - Goldberger, Assaf

AU - Strassler, Yossi

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2022/2

Y1 - 2022/2

N2 - A (classical) partial Hadamard Matrix is an m× n matrix H with values in { - 1 , 1 } such that HHT= nIm. 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 XXT= 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.

AB - A (classical) partial Hadamard Matrix is an m× n matrix H with values in { - 1 , 1 } such that HHT= nIm. 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 XXT= 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.

KW - Gram factorization problem

KW - Hadamard matrices

KW - Lattice basis reduction

UR - http://www.scopus.com/inward/record.url?scp=85118610850&partnerID=8YFLogxK

U2 - 10.1007/s10801-021-01077-z

DO - 10.1007/s10801-021-01077-z

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AN - SCOPUS:85118610850

SN - 0925-9899

VL - 55

SP - 217

EP - 244

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

IS - 1

ER -