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Asymptotically optimal approximate Hadamard matrices

Document Type

Article

Publication Date

10-22-2026

Abstract

An approximate Hadamard matrix is a well‐conditioned square matrix with all entries in . We measure the quality of a matrix by its condition number, that is, the ratio of its largest and smallest singular values. We prove that for any fixed positive , every sufficiently large dimension admits an approximate Hadamard matrix with condition number at most . In particular, the smallest possible condition number tends to  as . Conversely, there exists an absolute constant  such that for every sufficiently large , every  matrix with entries in  has condition number at least . Along the way, we resolve a problem of Jaming and Matolcsi concerning flat orthogonal matrices, and we conclude by describing several explicit infinite families of approximate Hadamard matrices.

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Funding note: NSF, Grant/Award Numbers: DMS2220320, DMS 2220304

Source Publication

Bulletin of the London Mathematical Society

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