Hadamard Matrices and Their Applications

An n × 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...

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Bibliographic Details
Published in:The Annals of statistics 1978-11, Vol.6 (6), p.1184-1238
Main Authors: Hedayat, A., Wallis, W. D.
Format: Article
Language:English
Subjects:
05-02
05B05
05B20
05B30
62-02
62K05
62K10
block design
Coding theory
Combinatorics
Determinants
Discrete mathematics
Equivalence relation
error correcting code
Factorial design
fractional factorial design
Hadamard matrix
Integers
Matrices
optimal design
orthogonal array
Random variables
Youden design
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Summary:An n × 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.
ISSN:0090-5364
2168-8966
DOI:10.1214/aos/1176344370