Hadamard Matrices and Their Applications
A. S. Hedayat,W. D. Wallis +1 more
TLDR
Hadamard matrices have been widely studied in the literature and many of their applications can be found in this paper, e.g., incomplete block designs, Youden designs, orthogonal $F$-square designs, optimal saturated resolution III (SRSIII), 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.Abstract:
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.read more
Citations
More filters
Journal ArticleDOI
G-optimal N-observation first order 2 k designs
TL;DR: The G-optimal first order (weighing) design with k factors and N observations is examined and it is shown that for N = 1 (mod 4), k < N - 1, the G -optimal, whenever a Hadamard matrix of order N -1 exists.
Journal ArticleDOI
On the Parameters of 2- Class Hadamard Association Schemes
S.N. Singh,Om Prakash Dubey +1 more
TL;DR: In this article, Singh and Dubey have proposed a technique for constructing H-matrices of the forms II and III of a 2-class association scheme from any 2-AS, whose parameters are given.
Journal ArticleDOI
On degree-d zero-sum sets of full rank
TL;DR: The smallest degree-d zero-sum sets of full rank are studied, i.e., those that contain n linearly independent elements, and relations to degree-1 annihilator spaces of Boolean functions and semi-orthogonal matrices are shown.
Journal ArticleDOI
Skellam mixture mechanism
Ergute Bao,Yizheng Zhu,Xiaokui Xiao,Yi Yang,Beng Chin Ooi,Benjamin Hong Meng Tan,Khin Mi Mi Aung +6 more
TL;DR: This paper focuses on the scenario where sensitive data are distributed among multiple participants, who jointly train a model through federated learning, using both secure multiparty computation to ensure the confidentiality of each gradient update, and differential privacy to avoid data leakage in the resulting model.
Journal ArticleDOI
The fractal nature of folds and the Walsh copolymers
TL;DR: The fractal nature of protein folds and some mathematical apparatus to describe them are discussed using semigroup theory and a special role is given to the Walsh functions and associated with them molecular constructions called Walsh copolymers.