One hundred years ago, in 1893, Jacques Hadamard [31] found square matrices of orders 12 and 20, with entries ±1, which had all their rows (and columns) pairwise orthogonal. These matrices, X = (Xij), satisfied the equality of the following inequality, |detX|2 ≤ ∏ ∑ |xij|2, and so had maximal determinant among matrices with entries ±1. Hadamard actually asked the question of finding the maximal determinant of matrices with entries on the unit disc, but his name has become associated with the question concerning real matrices. Disciplines Physical Sciences and Mathematics Publication Details Jennifer Seberry and Mieko Yamada, Hadamard matrices, Sequences, and Block Designs, Contemporary Design Theory – A Collection of Surveys, (D. J. Stinson and J. Dinitz, Eds.)), John Wiley and Sons, (1992), 431-560. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/1070 11 "-. Hadamard Matrices, Sequences, and Block Designs Jennifer Seberry and Mieko Yamada 1 IN1RODUCTION 2 HADAMARD MATRICES 3 THE SmONGEST HADAMARD CONSmUCTION THEOREMS 4 ORTIIOGONAL DESIGNS AND AsYMPTOTIC EXISTENCE 5 SEQUENCES 6 AMICABLE HADAMARD MAmICES AND AOD 7 CoNSmUCTIONS FOR SKEW HADAMARD MAmICES 8 M -SmucTUREs 9 WILLIAMSON AND WILUAMSON-TYPE MAmICES 10 SBIBD AND THE EXCESS OF HADAMARD MATRICES 11 CoMPLEX HADAMARD MATRICES APPENDIX REFERENCES

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Hadamard matrices, Sequences, and Block Designs

The first equality in (1.1) is a simple consequence of the fact that the sequence {n}9 1 < n < p — 1, runs through the set of kth power residues (mod/?) exactly k times. The primary purpose of this paper is to survey the present knowledge on the values of the Gauss sums §(k) and G(x), and to convey some of the principal ideas used in their determinations. We also briefly discuss more general Gauss sums. We begin by making some elementary remarks about the values of Gauss sums. It is easily verified by direct multiplication that, for nonprincipal x>

https://math.ucsd.edu/~revans/Determination.pdf

The determination of Gauss sums

https://www.cse.ust.hk/faculty/cding/JOURNALS/ffa98.pdf

New Generalized Cyclotomy and Its Applications

On Orthogonal Matrices Majid Behbahani Depar tment of Mathemat i c s and Computer Science Universi ty of Lethbridge M. Sc. Thesis , 2004 Our main aim in this thesis is to study and search for orthogonal matrices which have a certain kind of block structure. The most desirable class of matrices for our purpose are orthogonal designs constructible from 16 circulant matrices. In studying these ma­ trices, we show that the OD(l2; 1,1,1, 9) is the only orthogonal design constructible from 16 circulant matrices of type OD(4n; 1,1, l , 4n — 3), whenever n > 1 is an odd integer. We then use an exhaustive search to show that the only orthogonal design con­ structible from 16 circulant matrices of order 12 on 4 variables is the OD(12; 1 ,1,1, 9). It is known that by using of T-matrices and orthogonal designs constructible from 16 circulant matrices one can produce an infinite family of orthogonal designs. To com­ plement our studies we reproduce an important recent construction of T-matrices by Xia and Xia. We then turn our attention to the applications of orthogonal matrices. In some recent works productive regular Hadamard matrices are used to construct many new infinite families of symmetric designs. We show that for each integer n for which 4n is the order of a Hadamard matrix and 8n 2 — 1 is a prime, there is a productive regular Hadamard matrix of order 16n 2 (8n 2 — l ) 2 . As a corollary, we get many new infinite classes of symmetric designs whenever either of 4n(8n 2 — 1) — 1, 4n(8n 2 — 1) + 1 is a prime power. We also review some other constructions of productive regular Hadamard matrices which are related to our work.

On orthogonal matrices

Linear Complexity of Generalized Cyclotomic Binary Sequences of Order 2

/pdf/a-family-of-difference-sets-1e531fgldx.pdf

A family of difference sets

/pdf/the-cyclotomic-numbers-of-order-twenty-q6cxgsfxlf.pdf

The cyclotomic numbers of order twenty

Theorems on Dedekind Sums

An infinite family of Hadamard matrices of Williamson type

We give a formulation, via (1, −1) matrices, of Mathon's construction for conference matrices and derive a new family of conference matrices of order 5⋅92t+1 + 1,t ≥ 0. This family produces a new conference matrix of order 3646 and a new Hadamard matrix of order 7292. In addition we construct new families of Hadamard matrices of orders 6⋅92t+1 + 2, 10⋅92t+1 + 2, 8⋅49⋅9
 t
 ,t ≥ 0;q
2(q + 3) + 2 whereq ≡ 3 (mod 4) is a prime power and 1/2(q + 5) is the order of a skew-Hadamard matrix); (q + 1)q
2⋅9
 t
 ,t ≥ 0 (whereq ≡ 7 (mod 8) is a prime power and 1/2(q + 1) is the order of an Hadamard matrix). We also give new constructions for Hadamard matrices of order 4⋅9
 t
 ≥ 0 and (q + 1)q
2 (whereq ≡ 3 (mod 4) is a prime power).

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Albert Leon Whiteman

Papers

A family of difference sets

The cyclotomic numbers of order twenty

Theorems on Dedekind Sums

An infinite family of Hadamard matrices of Williamson type

New Hadamard matrices and conference matrices obtained via Mathon's construction