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arXiv · 2402.13202

A Note on Approximate Hadamard Matrices

Stefan Steinerberger
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Abstract

A Hadamard matrix is a scaled orthogonal matrix with $\pm 1$ entries. Such matrices exist in certain dimensions: the Hadamard conjecture is that such a matrix always exists when $n$ is a multiple of 4. A conjecture attributed to Ryser is that no circulant Hadamard matrices exist when $n > 4$. Recently, Dong and Rudelson proved the existence of approximate Hadamard matrices in all dimensions: there exist universal $0< c < C < \infty$ so that for all $n \geq 1$, there is a matrix $A \in \left\{-1,1\right\}^{n \times n}$ satisfying, for all $x \in \mathbb{R}^n$, $$ c \sqrt{n} \|x\|_2 \leq \|Ax\|_2 \leq C \sqrt{n} \|x\|_2.$$ We observe that, as a consequence of the existence of flat Littlewood polynomials, circulant approximate Hadamard matrices exist for all $n \geq 1$.

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Stefan Steinerberger. 2024-02-20. A Note on Approximate Hadamard Matrices. https://arxiv.org/abs/2402.13202

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