Physical Sciences and Mathematics Commons^™

Semi Williamson Type Matrices And The W(2n, N) Conjecture, Jennifer Seberry, Xian-Mo Zhang

Faculty of Informatics - Papers (Archive)

Four (1, -1, 0)-matrices of order m, X = (Xij), Y = (Yij), Z = (Zij), U = (Uij) satisfying

(i) XX^T + yyT + ZZ^T + UU^T = 2mI_m ,

(ii) x²_ij + y²_ij + z²_ij + U²_ij = 2, i, j = 1, ... ,m,

(iii) X, Y, Z, U mutually amicable,

will be called semi Williamson type matrices of order m. In this paper we prove that if there exist Williamson type matrices of order n₁,...n_k. then there exist semi Williamson …

Selected Papers In Combinatorics - A Volume Dedicated To R.G. Stanton, Jennifer Seberry, Brendan Mckay, Scott Vanstone

Faculty of Informatics - Papers (Archive)

Professor Stanton has had a very illustrious career. His contributions to mathematics are varied and numerous. He has not only contributed to the mathematical literature as a prominent researcher but has fostered mathematics through his teaching and guidance of young people, his organizational skills and his publishing expertise. The following briefly addresses some of the areas where Ralph Stanton has made major contributions.

Product Of Four Hadamard Matrices, R. Craigen, Jennifer Seberry, Xian-Mo Zhang

Faculty of Informatics - Papers (Archive)

We prove that if there exist Hadamard matrices of order 4m, 4n, 4p, and 4q then there exists an Hadamard matrix of order 16mnpq. This improves and extends the known result of Agayan that there exists a Hadamard matrix of order 8mn if there exist Hadamard matrices of order 4m and 4n.

On Small Defining Sets For Some Sbibd(4t-1, 2t-1, T-1), Jennifer Seberry

Faculty of Informatics - Papers (Archive)

We conjecture that 2t - 1 specified sets of 2t - 1 elements are enough to define an SBIBD(4t - 1, 2t - 1, t - 1) when 4t - 1 is a prime or product of twin primes (corrigendum 6:62,1992). This means that in these cases 2t - 1 rows are enough to uniquely define the Hadamard matrix of order 4t.

We show that the 2t -1 specified sets can be used to first find the residual BIBD(2t,4t - 2, 2t - 1, t, t - 1) for 4t - 1 prime. This can then be uniquely used to …

Resolvable Designs Applicable To Cryptographic Authentication Schemes, Keith M. Martin, Jennifer Seberry, Peter Wild

Faculty of Informatics - Papers (Archive)

We consider certain resolvable designs which have application to doubly perfect cartesian authentication schemes. These generalise structures determined by sets of mutually orthogonal latin squares and are related to semi-latin squares and other designs which find application in the design of experiments.

Constructing Hadamard Matrices From Orthogonal Designs, Christos Koukouvinos, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

The Hadamard conjecture is that Hadamard matrices exist for all orders 1,2, 4t where t ≥ 1 is an integer. We have obtained the following results which strongly support the conjecture:

(i) Given any natural number q, there exists an Hadamard matrix of order 2^sq for every s ≥ [2log₂(q - 3].

(ii) Given any natural number q, there exists a regular symmetric Hadamard matrix with constant diagonal of order 2^2s q2 for s as before.

A significant step towards proving the Hadamard conjecture would be proving "Given any natural number q and constant C …

A Cubic Rsa Code Equivalent To Factorization, John Loxton, David S.P. Khoo, Gregory J. Bird, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

The RSA public-key encryption system of Rivest, Shamir, and Adelman can be broken if the modulus, R say, can be factorized. However, it is still not known if this system can be broken without factorizing R. A version of the RSA scheme is presented with encryption exponent ℓ ≡ 3 (mod 6). For this modified version, the equivalence of decryption and factorization of R can be demonstrated.

‘”Suggestions For Presentation Of A Twenty-Minute Talk”’, Dinesh Sarvate, Jennifer Seberry

Faculty of Informatics - Papers (Archive)

How to present a research talk which will be remembered for a long(!) time for its presentation and clarity is a question which every newcomer would like to ask. Yes, one may also have a slight itch to prove that one is working on a very hard problem and his/her solution is straight from "the BOOK". We would like to suggest some points to ponder on the presentation only (these points may also help reduce itching).

Hadamard Matrices, Sequences, And Block Designs, Jennifer Seberry, Mieko Yamada

Faculty of Informatics - Papers (Archive)

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 = (X_ij), satisfied the equality of the following inequality,

|detX|² ≤ ∏ ∑ |x_ij|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.

On Small Defining Sets For Some Sbibd(4t - 1, 2t - 1, T - 1), Jennifer Seberry

Faculty of Informatics - Papers (Archive)

We conjecture that 2t - 1 specified sets of 2t - 1 elements are enough to define an SBIBD(4t - 1,2t - 1, t - 1) when 4t - 1 is a prime or product of twin primes. This means that in these cases 2t - 1 rows are enough to uniquely define the Hadamard matrix of order 4t. We show that the 2t - 1 specified sets can be used to first find the residual BIBD(2t, 4t - 2, 2t - 1, t, t - 1) for 4t - 1 prime. This can then be uniquely used to complete …