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Articles 1981 - 2010 of 2013
Full-Text Articles in Physical Sciences and Mathematics
Families Of Weighing Matrices, Anthony V. Geramita, Norman J. Pullman, Jennifer Seberry
Families Of Weighing Matrices, Anthony V. Geramita, Norman J. Pullman, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
A weighing matrix is an n x n matrix W = W(n, k) with entries from {0, 1, -l}, satisfying WWt = kIn. We shall call k the degree of W. It has been conjectured that if n = 0 (mod 4) then there exist n x n weighing matrices of every degree k < n.
We prove the conjecture when n is a power of 2. If n is not a power of two we find an integer t < n for which there are weighing matrices of every degree
A Note On Supplementary Difference Sets, Jennifer Seberry
A Note On Supplementary Difference Sets, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
Let S1, S2,···, Sn be subsets of G, a finite abelian group of order v, containing k1, k2,...,kn elements respectively. Write Ti for the totality of all differences between elements of Si (with repetitions), and T for the totality of elements of all the Ti. We will denote this by T= T1 & T2 & ... & Tn. If T contains each non-zero element of G a fixed number of times, lambda say, then the sets S1, S2, ..., …
Orthogonal Designs Iii: Weighing Matrices, Anthony V. Geramita, Jennifer Seberry
Orthogonal Designs Iii: Weighing Matrices, Anthony V. Geramita, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
A weighing matrix W = W(n,k) of order n and weight k is a square (0,l,-l)-matrix satisfying
WWt -kIn
An orthogonal design of order n on a single variable is a weighing matrix and consequently the study of orthogonal designs is intimately connected with the study of weighing matrices.
This paper reviews and updates the current status of the conjectures:
I. Let n = 2 (mod 4). Then there exists a W(n,k) if and only if k < n - 1 is the sum of two integer squares;
II. Let n = 0 (mod 4). Then there exists a W(n,k) for each k < n. This conjecture has been verified for n = 28, 2t+l, 2t +l·3 …
Williamson Matrices Of Even Order, Jennifer Seberry
Williamson Matrices Of Even Order, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and Williamson-type matrices. These latter are four (1,-1) matrices A,B,C,D, of order m, which pairwise satisfy
(i) MNT = NMT, M,N E (A,B,C,D),
and (ii) AAT +BBT +CCT +DDT = 4mIm, where I is the identity matrix.
Currently Williamson matrices are known to exist for all orders less than 100 except: 35,39,47,53,59,65,67,70,71,73,76,77,83,89,94.
This paper gives two constructions for Williamson matrices of even order, 2n. This is most significant when no Williamson matrices of order n …
A Survey Of Orthogonal Designs, Anthony V. Geramita, Jennifer Seberry
A Survey Of Orthogonal Designs, Anthony V. Geramita, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
This paper surveys orthogonal designs which are an overview of Baumert-Hall arrays, Hadamard matrices and weighing matrices.
The known results are given and unsolved problems indicated.
Kronecker Products And Bibds, Jennifer Seberry
Kronecker Products And Bibds, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
Recursive constructions are given which permit, under conditions described in the paper, a (v, b, r, k, lambda)-configuration to be used to obtain a (v', b', r', k, lambda)-configuration.
Although there are many equivalent definitions we will mean by a (v, b, r, k, lambda)-configuration or BIBD that (0, 1)-matrix A of size v x b with row sum r and column sum k satisfying
AAT = (r - lambda)I + lambdaJ
where, as throughout the remainder of this paper, I is the identity matrix and J the matrix with every element +1 whose sizes should be determined from …
A Note On Amicable Hadamard Matrices, Jennifer Seberry
A Note On Amicable Hadamard Matrices, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
The existence of Szekeres difference sets, X and Y, of size 2f with y E Y = -y E Y, where q = 4f + 1 is a prime power, q = 5 (mod 8) and q = p2 + 4, is demonstrated. This gives amicable Hadamard matrices of order 2(q + 1), and if 2q is also the order of a symmetric conference matrix, a regular symmetric Hadamard matrix of order 4q2 with constant diagonal.
Hadamard Matrices Of Order 28m, 36m, And 44m, Jennifer Seberry
Hadamard Matrices Of Order 28m, 36m, And 44m, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
We show that if four suitable matrices of order m exist then there are Hadamard matrices of order 28 m, 36 m, and 44 m. In particular we show that Hadamard matrices of orders 14(q + 1), 18(q + 1), and 22(q + 1) exist when q is a prime power and q = l(mod 4).
Also we show that if n is the order of a conference matrix there is an Hadamard matrix of order 4mn. As a consequence there are Hadamard matrices of the following orders less than 4000:
476, 532, 836, 1036, 1012, 1100, 1148, 1276, 1364, …
Recent Advances In The Construction Of Hadamard Matrices, Jennifer Seberry
Recent Advances In The Construction Of Hadamard Matrices, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
In the past few years exciting new discoveries have been made in constructing Hadamard matrices. These discoveries have been centred in two ideas:
(i) the construction of Baumert-Hall arrays by utilizing a construction of L. R. Welch, and
(ii) finding suitable matrices to put into these arrays.
We discuss these results, many of which are due to Richard J. Turyn or the author.
A List Of Balanced Incomplete Block Designs For R < 30, Jane W. Di Paola, Jennifer Seberry, W D. Wallis
A List Of Balanced Incomplete Block Designs For R < 30, Jane W. Di Paola, Jennifer Seberry, W D. Wallis
Faculty of Informatics - Papers (Archive)
A balanced incomplete block design consists of a set of v elements arranged into b k-element subsets called blocks such that each element occurs r times and each pair of elements appears in lambda distinct blocks. The numbers v,b,r,k,lambda are called the parameters of the design. A necessary condition that a design exist is that the parameters be integers satisfying:
(1) vr = bk
( 2) r(k-1) = lambda (v-1)
Families Of Codes From Orthogonal (0,1,-1)-Matrices, Jennifer Seberry
Families Of Codes From Orthogonal (0,1,-1)-Matrices, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
Sloane and Seidel have constructed (n,2n,1/2(n-2)) and (n-1,2n,1/2(n-2)) codes whenever n = 1 + a2 + b2 = 2(mod 4), a,b integer, is the order of a conference matrix. We give constructions for (n,2n,1/2(n-2)) and (n-1,2n,1/2(n-4)) codes when n = 2(mod 4) and conference matrices cannot exist.
In particular we give results for n = 22, 34, 66, 70, 106,130,154,162,202,210, ... ,"210, ... , but our codes are not as ""good" as those from Hadamard matrices or of Sloane and Seidel".
Some Matrices Of Williamson Type, Jennifer Seberry
Some Matrices Of Williamson Type, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and four (1,-1) matrices A, B, C, D of order m which are of Williamson type; that is, they pairwise satisfy
(i) MNT = NMT, and
(ii) AAT + BBT + CCT + DDT = 4mIm
We show that if p = 1 (mod 4) is a prime power then such matrices exist for m = 1/2p(p+1). The matrices constructed are not circulant and need not be symmetric. This means there are Hadamard …
A Note On Bibds, Jennifer Seberry
A Note On Bibds, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
A balanced incomplete block design or BlBD is defined as an arrangement of v objects in b blocks, each block containing k objects all different, so that there are r blocks containing a given object and lambda blocks containing any two given objects.
In this note we shall extend a method of Sprott [2, 3] to obtain several new families of BIBD's. The method is based on the first Module Theorem of Bose [1] for pure differences.
We shall frequently be concerned with collections in which repeated elements are counted multiply, rather than with sets. If T1 and T …
Complex Hadamard Matrices, Jennifer Seberry
Complex Hadamard Matrices, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
R. J. Turyn introduced complex Hadamard matrices and showed that if there is a complex Hadamard matrix of order c and a real Hadamard matrix of order h > 1, then there is a real Hadamard matrix of order hc.
Previously, complex Hadamard matrices were only known for a few small orders and the orders for which symmetric conference matrices were known. These latter are known only to exist for orders which can be written as 1 + a2 + b2 where a, b are integers.
We give many constructions for new infinite classes of complex Hadamard matrices and …
Some Remarks On Supplementary Difference Sets, Jennifer Seberry
Some Remarks On Supplementary Difference Sets, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
Let S1,S2 ,... ,Sn be subsets of V, a finite abelian group of order v written in additive notation, containing k1 k2,... ,kn elements respectively. Write Ti for the totality of all differences between elements of Si (with repetitions), and T for the totality of elements of all the Ti. If T contains each non-zero element of V a fixed number of times, lambda say, then the sets S1, S2,... ,Sn will be called n - {v; k1, k2, …
Some Classes Of Hadamard Matrices With Constant Diagonal, Jennifer Seberry, Albert Leon Whiteman
Some Classes Of Hadamard Matrices With Constant Diagonal, Jennifer Seberry, Albert Leon Whiteman
Faculty of Informatics - Papers (Archive)
The concepts of circulant and back circulant matrices are generalized to obtain incidence matrices of subsets of finite additive abelian groups. These results are then used to show the existence of skew-Hadamard matrices of order 8(4f+l) when f is odd and 8f + 1 is a prime power. This shows the existence of skew-Hadamard matrices of orders 296, 592, 1184, 1640, 2280, 2368 which were previously unknown.
A Construction For Hadamard Arrays, Joan Cooper, Jennifer Seberry
A Construction For Hadamard Arrays, Joan Cooper, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
We give a construction for Hadamard arrays and exhibit the arrays of orders 4t , tE{l,3,5,7, ... 19} This gives seventeen new Hadamard matrices of order less than 4000.
Orthogonal (0,1,-1) Matrices, Jennifer Seberry
Orthogonal (0,1,-1) Matrices, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
We study the conjecture: There exists a square (0,l,-l)-matrix W = W(w,k) of order w satisfying
WWT= kIw
for all k = 0, 1,..., w when w = 0 (mod 4). We prove the conjecture is true for 4, 8, 12, 16, 20, 24, 28, 32, 40 and give partial results for 36, 44, 52, 56.
On Integer Matrices Obeying Certain Matrix Equations, Jennifer Seberry
On Integer Matrices Obeying Certain Matrix Equations, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
We discuss integer matrices B of odd order v which satisfy
Br = ± B, BBr = vI - J, BJ = O.
Matrices of this kind which have zero diagonal and other elements ± 1 give rise to skew-Hadamard and n-type matrices; we show that the existence of a skew-Hadamard (n-type) matrix of order h implies the existence of skew-Hadamard (n-type) matrices of orders (h - 1)5 + 1 and (h - 1)7 + 1. Finally we show that, although there are matrices B with elements other than ± 1 and 0, the equations force considerable restrictions …
Cyclotomy, Hadamard Arrays And Supplementary Difference Sets, David C. Hunt, Jennifer Seberry
Cyclotomy, Hadamard Arrays And Supplementary Difference Sets, David C. Hunt, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
A 4n x 4n Hadamard array, H, is a square matrix of order 4n with elements ± A, ± B, ± C, ± D each repeated n times in each row and column. Assuming the indeterminates A, B, C, D commute, the row vectors of H must be orthogonal. These arrays have been found for n = 1 (Williamson, 1944), n = 3 (Baumert-Hall, 1965), n = 5 (Welch, 1971), and some other odd n < 43 (Cooper, Hunt, Wallis).
The results for n = 25, 31, 37, 41 are presented here, as is a result for n = 9 not based on supplementary difference …
On Supplementary Difference Sets, Jennifer Seberry
On Supplementary Difference Sets, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
Given a finite abelian group V and subsets S1, S2, ... ,Sn of V, write Ti for the totality of all the possible differences between elements of Si (with repetitions counted multiply) and T for the totality of members of all the Ti. If T contains each non-zero element of V the same number of times, then the sets S1, S2,...,Sn will be called supplementary difference sets.
We discuss some properties for such sets, give some existence theorems and observe their use in the construction of Hadamard …
A Skew-Hadamard Matrix Of Order 92, Jennifer Seberry
A Skew-Hadamard Matrix Of Order 92, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
Previously the smallest order for which a skew-Hadamard matrix was not known was 92. We construct such a matrix below.
Some (1, -1) Matrices, Jennifer Seberry
Some (1, -1) Matrices, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
We define an n-type (1, -1) matrix N = 1 + R of order n ~ 2 (mod 4) to have R symmetric and R2 = (n - 1)/n. These matrices are analogous to skewtype matrices M = 1 + W which have W skew-symmetric.
Amicable Hadamard Matrices, Jennifer Seberry
Amicable Hadamard Matrices, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
If X is a symmetric Hadamard matrix, Y is a skew-Hadamard matrix, and XYT is symmetric, then X and Y are said to be amicable Hadamard matrices. A construction for amicable Hadamard matrices is given, and then amicable Hadamard matrices are used to generalize a construction for skew-Hadamard matrices.
Combinatorial Matrices, Jennifer Seberry
Combinatorial Matrices, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
We investigate the existence of integer matrices B satisfying the equation BBT = rI + sJ where T denotes transpose, r and s are integers, I is the identity matrix and J is the matrix with every element +1.
Some Results On Configurations, Jennifer Seberry
Some Results On Configurations, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
A (v, k, lambda) configuration is conjectured to exist for every v, k and lambda satisfying lambda(v-l) = k(k-l) and k - lambda is a square if v is even, x2 = (k - lambda)y2+(-1)(v-1)/2lamdaZ2 has a solution in integers x,y and z not all zero for v odd.
Integer Matrices Obeying Generalized Incidence Equations, Jennifer Seberry
Integer Matrices Obeying Generalized Incidence Equations, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
We consider integer matrices obeying certain generalizations of the incidence equations for (v, k, lambda)-configurations and show that given certain other constraints, a constant multiple of the incidence matrix of a (v, k, lambda)-configuration may be identified as the solution of the equation.
(V, K, Lambda)-Configurations And Hadamard Matrices, Jennifer Seberry
(V, K, Lambda)-Configurations And Hadamard Matrices, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
(v, k, lambda) Configirations and Hadamard matrics
Hadamard Designs, Jennifer Seberry
Hadamard Designs, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
In this paper it is shown that an Hadamard design with each letter repeated once and only once can exist for 2, 4 and 8 letters only. L.D. Baumert and Marshall Hall, Jr have found a design with four letters each repeated three times. Their design and the design on four letters each repeated once, found by J. Williamson, is the totality previously published.
A Note Of A Class Of Hadamard Matrices, Jennifer Seberry
A Note Of A Class Of Hadamard Matrices, Jennifer Seberry
Faculty of Informatics - Papers (Archive)
An Hadamard matrix H is a matrix of order n all of whose elements are + 1 or -1 and which satisfies H ffT = nIn . H = S + In is a skew-type Hadamard matrix if ST = -So It is conjectured that an Hadamard matrix always exists for n = 4t, t any integer. Many known matrices and classes of matrices can be found in [1].