The Annals of Statistics

Hadamard Matrices and Their Applications

A. Hedayat and W. D. Wallis

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An $n \times n$ matrix $H$ with all its entries $+1$ and $-1$ is Hadamard if $HH' = nI$. It is well known that $n$ must be 1, 2 or a multiple of 4 for such a matrix to exist, but is not known whether Hadamard matrices exist for every $n$ which is a multiple of 4. The smallest order for which a Hadamard matrix has not been constructed is (as of 1977) 268. Research in the area of Hadamard matrices and their applications has steadily and rapidly grown, especially during the last three decades. These matrices can be transformed to produce incomplete block designs, $t$-designs, Youden designs, orthogonal $F$-square designs, optimal saturated resolution III designs, optimal weighing designs, maximal sets of pairwise independent random variables with uniform measure, error correcting and detecting codes, Walsh functions, and other mathematical and statistical objects. In this paper we survey the existence of Hadamard matrices and many of their applications.

Article information

Ann. Statist., Volume 6, Number 6 (1978), 1184-1238.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 05B20: Matrices (incidence, Hadamard, etc.)
Secondary: 62K05: Optimal designs 62K10: Block designs 05B05: Block designs [See also 51E05, 62K10] 05B30: Other designs, configurations [See also 51E30]

62-02 05-02 Hadamard matrix block design optimal design fractional factorial design orthogonal array error correcting code Youden design


Hedayat, A.; Wallis, W. D. Hadamard Matrices and Their Applications. Ann. Statist. 6 (1978), no. 6, 1184--1238. doi:10.1214/aos/1176344370.

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