Skew Hadamard difference sets from the Ree-Tits slice symplectic spreads in PG(3,3^{2h+1})

Using a class of permutation polynomials of $F_{3^{2h+1}}$ obtained from the Ree-Tits symplectic spreads in $PG(3,3^{2h+1})$, we construct a family of skew Hadamard difference sets in the additive group of $F_{3^{2h+1}}$. With the help of a computer, we show that these skew Hadamard difference sets are new when $h=2$ and $h=3$. We conjecture that they are always new when $h>3$. Furthermore, we present a variation of the classical construction of the twin prime power difference sets, and show that inequivalent skew Hadamard difference sets lead to inequivalent difference sets with twin prime power parameters.Comment: 18 page

On quaternary complex Hadamard matrices of small orders

One of the main goals of design theory is to classify, characterize and count various combinatorial objects with some prescribed properties. In most cases, however, one quickly encounters a combinatorial explosion and even if the complete enumeration of the objects is possible, there is no apparent way how to study them in details, store them efficiently, or generate a particular one rapidly. In this paper we propose a novel method to deal with these difficulties, and illustrate it by presenting the classification of quaternary complex Hadamard matrices up to order 8. The obtained matrices are members of only a handful of parametric families, and each inequivalent matrix, up to transposition, can be identified through its fingerprint.Comment: 7 page

THE PROPERTIES OF SOME GOODNESS-OF-FIT TESTS

The properties of Pearson’s goodness-of-fit test, as used in density forecast evaluation, income distribution analysis and elsewhere, are analysed. The components-of-chi-squared or “Pearson analog” tests of Anderson (1994) are shown to be less generally applicable than was originally claimed. For the case of equiprobable classes, where the general components tests remain valid, a Monte Carlo study shows that tests directed towards skewness and kurtosis may have low power, due to differences between the class boundaries and the intersection points of the distributions being compared. The power of individual component tests can be increased by the use of nonequiprobable classes.Pearson’s Goodness-of-fit test ; Component tests ; Distributional assumptions ; Monte Carlo ; Normality ; Nonequiprobable partitions

Sedenion extension loops and frames of hypercomplex 2[superscript n]-ons

This research work is mainly on hypercomplex numbers with major emphasis on the 16 dimensional sedenions. In Chapter 1, a literature review on current and past work on the area of hypercomplex numbers is given. Basic definitions to be used in the rest of the work are given. We introduce the sedenions and the different doubling formulas for obtaining sedenions from the octonions and general 2n-ons from 2n -1-ons. We also discuss some advantages and disadvantages of these doubling formulas.;In Chapter 2 the sedenions are considered from the point of view of loop theory. Although the sedenions do not directly yield loops in the way that Moufang loops are obtained from octonions, the left loop of non-zero sedenions does contain new two-sided subloops as sedenion extensions. These loops are constructed abstractly as extensions of subloops L of the octonions. The chapter examines the satisfaction by these extensions of various standard loop-theoretical identities. Equivalent conditions for the extensions to be groups are discussed. Consequences arising from these conditions are also given. For example, it turns out that they are all power-associative, even though the full left loop of all non-zero sedenions is not itself power-associative.;Chapter 3 gives a powerful result, the multiplication of the frames of the 2n-ons fits in the projective geometry PG(n - 1, 2). In doing so, it is convenient to consider this projective geometry from several different viewpoints: as a design, in terms of Nim addition, and as the geometry of linear subspaces of a vector space. It is then shown that the number of nontrivial subloops of order 2k inside the loop formed by the frame of the 2n-ons, the set of basic elements together with their negatives, is given by nk 2=2n-1 2n-1-1 2n-2-1&cdots;2 n-k+1-12 1-122-1 23-1&cdots; 2k-1.;In Chapter 4, Hadamard matrices are discussed. It is shown that the sign matrix of the frame multiplication in the 2n-ons under the Smith-Conway or Cayley-Dickson process is a skew Hadamard matrix. These matrices are shown to be equivalent to Kronecker products when n ≤ 3. Chapter 5 gives some open problems for future research