Journal Article

On The Excess of Hadamard Matrix

Leghwel, Abd Alrzak M, Lashhab, Mohammed I

2011

Overview

A (-1, 1) -matrix is a matrix whose only entries are the numbers -1 or 1. In this paper for the most part we will be interested in special (-1,1) -matrices called Hadamard matrices. A Hadamard matrix of order n is an n x n (-1,1) - matrix H, satisfying H'H = H'H = nIn, where H' denotes the transpose of H and In is the identity matrix of order n. If H is a Hadamard matrix of order n, let w(H) = number of plus ones in H and let σ (H) = sum of all the entries of H. The numbers w(H) and σ (H) are called the weight of H and the excess of H, respectively. Further let w(n)= max { w(H) :H EΩ (n) } and σ (n)= max { σ (H) : H EΩ(n) } where Ω(n) is the class of all Hadamard matrices of order n. We call w(n) and σ (n) the maximum weight and the maximum excess of the class Q(n), respectively. The functions w and a were first introduced by Schmidt (1973) and subsequently studied by Schmidt and Wang (1977), Best (1977), Enomoto and Miyamoto (1980), Hammer, Levingston, Seberry (1978), and many other authors. The purpose of this paper is to report on what progress has been made on the maximum excess problem or equivalently the problem of maximum weight. In this paper, we first derive the relationship between w(H) and σ (H) as well as between w(n) and σ (n). The paper then proceeds to elaborate on the papers by Schmidt and Wang (1977) and Best (1977). Perhaps a key and most useful result in this paper is the inequality σ (n) ≤ n√n obtained by Best (1977). We conclude this paper by giving some results and examples of Hadamard matrices with maximum excess.

• Publisher: الجامعة الأسمرية الإسلامية زليتن - كليتى الآداب والعلوم
• Subject: الكيمياء العضوية / المركبات الحيوية / مصفوفة هدامارد