A note on approximate Hadamard matrices

IF 1.4 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Designs, Codes and Cryptography Pub Date : 2024-06-04 DOI:10.1007/s10623-024-01430-w

Stefan Steinerberger

引用次数: 0

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 $n \ge 1$, there is a matrix $A \in \left\{ -1,1\right\} ^{n \times n}$ satisfying, for all $x \in \mathbb {R}^n$,

$$\begin{aligned} c \sqrt{n} \Vert x\Vert _2 \le \Vert Ax\Vert _2 \le C \sqrt{n} \Vert x\Vert _2. \end{aligned}$$

We observe that, as a consequence of the existence of flat Littlewood polynomials, circulant approximate Hadamard matrices exist for all $n \ge 1$.

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关于近似哈达玛矩阵的说明

哈达玛矩阵是一个具有 $\pm 1$ 条目的按比例正交矩阵。这样的矩阵存在于某些维度中：哈达玛猜想是，当 n 是 4 的倍数时，这样的矩阵总是存在的。雷塞尔提出的一个猜想是，当 $n > 4$ 时，不存在环形哈达玛矩阵。最近，Dong 和 Rudelson 证明了所有维度上近似 Hadamard 矩阵的存在：存在普遍的（0< c< C<）矩阵，这样对于所有的（n），都有一个矩阵（A）满足，对于所有的（x），$$\begin{aligned} c \sqrt{n}\Vert x\Vert _2 \le \Vert Ax\Vert _2 \le C \sqrt{n}\Vert x\Vert _2.\end{aligned}$$我们注意到，由于平利特尔伍德多项式的存在，对于所有的 $n \ge 1$ 都存在环形近似哈达玛矩阵。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.