On the number of Hadamard matrices via anti-concentration

Asaf Ferber,Vishesh Jain,Yufei Zhao

doi:10.1017/s0963548321000377

Abstract

AbstractMany 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. $\{\pm 1\}$) 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 $n\times n$ Hadamard matrices and (ii) we improve a recent bound of Deneanu and Vu on the probability of normality of a random $\{\pm 1\}$ matrix.

Highlights

The number of Hadamard matrices A square matrix H of order n whose entries are {±1} is called a Hadamard matrix of order n if its rows are pairwise orthogonal, i.e. if HHT = nIn. These matrices are named after Jacques Hadamard, who studied them in connection with his maximal determinant problem
Hadamard asked for the maximum value of the determinant of any n × n square matrix all of whose entries are bounded in absolute value by 1
We refer the reader to the surveys [HW+78, SY92] and the books [Aga06, Hor12] for a comprehensive account of Hadamard matrices and their numerous applications

Summary

Introduction

(ii) The {±1}n vectors spanning a subspace in our case are highly dependent due to the mutual orthogonality constraint – as the proof of the trivial upper bound at the start of the subsection shows, the probability that the rows of a random k × n {±1} matrix are mutually orthogonal is 2− (k2)√; this rules out the strategy of conditioning on the rows being orthogonal when k = ( n), even if one were to prove a variant of the result of Kahn, Komlós and Szemerédi to deal with orthogonal complements. To deal with (i), we will show that for any k × n matrix A which has this linear algebraic structure, the number of solutions x in {±1}n to Ax = 0 is at most 2n−(1+C)k, where C > 0 is a constant depending only on c1 and c2 Using these improved bounds with the same strategy as for the trivial proof, we see that for n sufficiently large, n−1.

Improved Halász-type inequalities

Anti-concentration

The replication trick