Related papers: Polynomial Criterion for Abelian Difference Sets
We provide some applications of a polynomial criterion for difference sets. These include counting the difference sets with specified parameters in terms of Hilbert functions, in particular a count of bent functions. We also consider the…
A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are…
A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group $G$. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over…
We describe general connections between intersective properties of sets in Abelian groups and positive exponential sums. In particular, given a set $A$ the maximal size of a set whose difference set avoids $A$ will be related to positive…
Let $G$ be a finite group acting on a finite dimensional complex vector space $V$ via linear transformations. Let $\mathbb{C}[V]^G$ be the algebra of polynomials that are invariant under the induced $G$-action on the polynomial ring…
We call a finite, spanning set of a semi-simple real Lie algebra a distinguished set if it satisfies the following property: The Lie bracket of any two elements out of the set is, up to some constant, another element in the set; conversely,…
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…
A separating algebra is, roughly speaking, a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a geometric notion of separating…
This paper deals with the comparison of two common types of equivalence groups of differential equations, and this gives rise to a number of results presented in the form of theorems. It is shown in particular that one type can be…
By definition, admissible matrix groups are those that give rise to a wavelet-type inversion formula. This paper investigates necessary and sufficient admissibility conditions for abelian matrix groups. We start out by deriving a block…
We define a notion of an arithmetic set in an arbitrary countable group and study properties of these sets in the cases of Abelian groups and non-abelian free groups.
Our main objective is to show that the computational methods that we previously developed to search for difference families in cyclic groups can be fully extended to the more general case of arbitrary finite abelian groups. In particular…
In this paper, we introduce a new function related to the sum of element orders of finite groups. It is used to give some criteria for a finite group to be cyclic, abelian, nilpotent, supersolvable and solvable, respectively.
The DifSets package for GAP implements an algorithm for enumerating all difference sets in a group up to equivalence and provides access to a library of results. The algorithm functions by finding difference sums, which are potential images…
A theory of matchings for finite subsets of an abelian group, introduced in connection with a conjecture of Wakeford on canonical forms for homogeneous polynomials, has since been extended to the setting of field extensions and to that of…
In general the endomorphisms of a non-abelian group do not form a ring under the operations of addition and composition of functions. Several papers have dealt with the ring of functions defined on a group which are endomorphisms when…
In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In…
We introduce a notion of dimension for the solution set of a system of algebraic difference equations that measures the degrees of freedom when determining a solution in the ring of sequences. This number need not be an integer, but, as we…
A matching in a group G is a bijection f from a subset A to a subset B in G such that af(a) does not belong to A for all a in A. The group G is said to have the matching property if, for any finite subsets A,B in G of same cardinality with…
In this paper we explore a connection between certain Almost Perfect Nonlinear Functions (APN functions) and relative difference sets. In particular, we show that the image set of certain 2-to-1 APN functions is a relative difference set.…