18 research outputs found
Greco-Latin squares as bijections
A Latin square of order n is an n-by-n array of n symbols, which we take to be the integers 0 to n-1, such that no symbol is repeated in any row or column. Two Latin squares of the same order are orthogonal if, when overlapped, no ordered pair of symbols occurs more than once. Equivalently, the Latin squares together form a bijection on the set of n-squared coordinates. In this thesis the question of what this bijection is in terms of projective planes is investigated. The major result here is a new necessary and sufficient condition such that two ternary rings correspond to the same plane
A new structure for difference matrices over abelian -groups
A difference matrix over a group is a discrete structure that is intimately
related to many other combinatorial designs, including mutually orthogonal
Latin squares, orthogonal arrays, and transversal designs. Interest in
constructing difference matrices over -groups has been renewed by the recent
discovery that these matrices can be used to construct large linking systems of
difference sets, which in turn provide examples of systems of linked symmetric
designs and association schemes. We survey the main constructive and
nonexistence results for difference matrices, beginning with a classical
construction based on the properties of a finite field. We then introduce the
concept of a contracted difference matrix, which generates a much larger
difference matrix. We show that several of the main constructive results for
difference matrices over abelian -groups can be substantially simplified and
extended using contracted difference matrices. In particular, we obtain new
linking systems of difference sets of size in infinite families of abelian
-groups, whereas previously the largest known size was .Comment: 27 pages. Discussion of new reference [LT04
PN functions, complete mappings and quasigroup difference sets
We investigate pairs of permutations of such that
is a permutation for every . We show that
necessarily for some complete mapping of
, and call the permutation a perfect nonlinear
(PN) function. If , then is a PcN function, which have
been considered in the literature, lately. With a binary operation on
involving , we obtain a
quasigroup, and show that the graph of a PN function is a difference
set in the respective quasigroup. We further point to variants of symmetric
designs obtained from such quasigroup difference sets. Finally, we analyze an
equivalence (naturally defined via the automorphism group of the respective
quasigroup) for PN functions, respectively, the difference sets in the
corresponding quasigroup
Good Random Matrices over Finite Fields
The random matrix uniformly distributed over the set of all m-by-n matrices
over a finite field plays an important role in many branches of information
theory. In this paper a generalization of this random matrix, called k-good
random matrices, is studied. It is shown that a k-good random m-by-n matrix
with a distribution of minimum support size is uniformly distributed over a
maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and
vice versa. Further examples of k-good random matrices are derived from
homogeneous weights on matrix modules. Several applications of k-good random
matrices are given, establishing links with some well-known combinatorial
problems. Finally, the related combinatorial concept of a k-dense set of m-by-n
matrices is studied, identifying such sets as blocking sets with respect to
(m-k)-dimensional flats in a certain m-by-n matrix geometry and determining
their minimum size in special cases.Comment: 25 pages, publishe
Some Proposed Problems on Permutation Polynomials over Finite Fields
From the 19th century, the theory of permutation polynomial over finite fields, that are arose in the work of Hermite and Dickson, has drawn general attention. Permutation polynomials over finite fields are an active area of research due to their rising applications in mathematics and engineering. The last three decades has seen rapid progress on the research on permutation polynomials due to their diverse applications in cryptography, coding theory, finite geometry, combinatorics and many more areas of mathematics and engineering. For this reason, the study of permutation polynomials is important nowadays. In this chapter, we propose some new problems in connection to permutation polynomials over finite fields by the help of prime numbers
Permutation polynomials and systems of permutation polynomials in several variables over finite rings
This paper will present the historical development of theorems regarding permutation polynomials in several variables over finite fields. Single variable permutation polynomials will be discussed since they are so important to the discussions which will follow. Theorems involving permutation polynomials and systems of permutation polynomials will also be considered. It will be shown that many of the interesting results obtained for finite fields can be generalized to finite rings
Connectivity of some Algebraically Defined Digraphs
Let p be a prime, e a positive integer, q = pe, and ��q denote the finite field of q elements. Let fi : ��2q → ��q be arbitrary functions, where 1 ≤ i ≤1, i and l are integers. The digraph D = D(q:f), where f = f ,..., f l): ��2q → ��lq, is defined as follows. The vertex of D is ��l+1q. There is an arc from a vertex x = (x1,...xl+1) to a vertex y = (y1,...yl+1) if xi + yi = f i-l(x1, y1) for all i, 2 ≤ i ≤ l + 1. In this paper we study the strong connectivity of D and completely describe its strong components. The digraphs D are directed analogues of some algebraically defined graphs, which have been studied extensively and have many applications