On the number of Hadamard matrices via anti-concentration
Name
on-the-number-of-hadamard-matrices-via-anti-concentration.pdf
Description
Published version
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441.47 KB
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Adobe PDF
Checksum (MD5)
0ea12550e9bf9f5ee144a387c06bca94
Author(s)
Ferber, Asaf
Jain, Vishesh
Zhao, Yufei
Date Issued
2022
Journal
Combinatorics Probability and Computing
Publisher
Cambridge University Press (CUP)
Citation
Ferber, Asaf, Jain, Vishesh and Zhao, Yufei. 2022. "On the number of Hadamard matrices via anti-concentration." Combinatorics Probability and Computing, 31 (3).
Version
Final published version
Abstract
Many problems in combinatorial linear algebra require upper bounds on the number of solutions to an underdetermined system of linear equations Ax=b , where the coordinates of the vector x are restricted to take values in some small subset (e.g. {±1} ) of the underlying field. The classical ways of bounding this quantity are to use either a rank bound observation due to Odlyzko or a vector anti-concentration inequality due to Halász. The former gives a stronger conclusion except when the number of equations is significantly smaller than the number of variables; even in such situations, the hypotheses of Halász’s inequality are quite hard to verify in practice. In this paper, using a novel approach to the anti-concentration problem for vector sums, we obtain new Halász-type inequalities that beat the Odlyzko bound even in settings where the number of equations is comparable to the number of variables. In addition to being stronger, our inequalities have hypotheses that are considerably easier to verify. We present two applications of our inequalities to combinatorial (random) matrix theory: (i) we obtain the first non-trivial upper bound on the number of n×n Hadamard matrices and (ii) we improve a recent bound of Deneanu and Vu on the probability of normality of a random {±1} matrix.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution 4.0 International license
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DOI of Published Version
https://doi.org/10.1017/S0963548321000377
