On a relation between a cyclic relative difference set associated with the quadratic extensions of a finite field and the Szekeres difference sets

COMBINATORICA ◽  
1988 ◽  
Vol 8 (2) ◽  
pp. 207-216 ◽  
Author(s):  
Mieko Yamada
Keyword(s):  
Finite Field ◽  
Difference Sets ◽  
Difference Set ◽  
Relative Difference ◽  
Relative Difference Set ◽  
Quadratic Extensions

Divisible Semiplanes, Arcs, and Relative Difference Sets

1987 ◽  
Vol 39 (4) ◽  
pp. 1001-1024 ◽  
Author(s):  
Dieter Jungnickel
Keyword(s):  
Projective Plane ◽  
Difference Sets ◽  
Difference Set ◽  
Relative Difference ◽  
List Type ◽  
Relative Difference Set ◽  
Large Collection ◽  
Baer Subplane ◽  
Relative Difference Sets ◽  
Definition Of

In this paper we shall be concerned with arcs of divisible semiplanes. With one exception, all known divisible semiplanes D (also called “elliptic” semiplanes) arise by omitting the empty set or a Baer subset from a projective plane Π, i.e., D = Π\S, where S is one of the following:(i) S is the empty set.(ii) S consists of a line L with all its points and a point p with all the lines through it.(iii) S is a Baer subplane of Π.We will introduce a definition of “arc” in divisible semiplanes; in the examples just mentioned, arcs of D will be arcs of Π that interact in a prescribed manner with the Baer subset S omitted. The precise definition (to be given in Section 2) is chosen in such a way that divisible semiplanes admitting an abelian Singer group (i.e., a group acting regularly on both points and lines) and then a relative difference set D will always contain a large collection of arcs related to D (to be precise, —D and all its translates will be arcs).

On Automorphism Groups of Divisible Designs

1982 ◽  
Vol 34 (2) ◽  
pp. 257-297 ◽  
Author(s):  
Dieter Jungnickel
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Difference Sets ◽  
Difference Set ◽  
Relative Difference ◽  
Block Designs ◽  
Divisible Design ◽  
Finite Projective Spaces ◽  
Group Divisible ◽  
Singer Group

A (group) divisible design is a tactical configuration for which the v points are split into m classes of n each, such that points have joining number λ (resp. λ2) if and only if they are in the same (resp. in different) classes. We are interested in such designs with a nice automorphism group. We first investigate divisible designs with equally many points and blocks admitting an automorphism group acting regularly on all points and on all blocks, i.e., with a Singer group (Singer [50] obtained the first result in this direction for the finite projective spaces).As in the case of block designs, one may expect a divisible design with a Singer group to be equivalent to some sort of difference set; as it turns out, one here obtains a generalisation of the relative difference sets of Butson and Elliott [11] and [20].

Relations Among Generalized Hadamard Matrices, Relative Difference Sets, and Maximal Length Linear Recurring Sequences

1963 ◽  
Vol 15 ◽  
pp. 42-48 ◽  
Author(s):  
A. T. Butson
Keyword(s):  
Incidence Matrix ◽  
Hadamard Matrix ◽  
Block Design ◽  
Difference Sets ◽  
Hadamard Matrices ◽  
Difference Set ◽  
Relative Difference ◽  
Block Designs ◽  
Linear Recurring Sequences ◽  
Balanced Incomplete Block

It was established in (5) that the existence of a Hadamard matrix of order 4t is equivalent to the existence of a symmetrical balanced incomplete block design with parameters v = 4t — 1, k = 2t — 1, and λ = t — 1. A block design is completely characterized by its so-called incidence matrix. The existence of a block design with parameters v, k, and λ such that the corresponding incidence matrix is cyclic was shown in (3) to be equivalent to the existence of a cyclic difference set with parameters v, k, and λ. For certain values of the parameters, Hadamard matrices, block designs, and difference sets do coexist.

scholarly journals On non-symmetric relative difference sets

2008 ◽  
Vol 37 (3) ◽  
pp. 427-435 ◽  
Author(s):  
Yutaka HIRAMINE
Keyword(s):  
Difference Sets ◽  
Relative Difference ◽  
Relative Difference Sets

Relative difference sets in semidirect products with an amalgamated subgroup

2005 ◽  
Vol 13 (3) ◽  
pp. 211-221 ◽  
Author(s):  
John C. Galati ◽  
Alain C. LeBel
Keyword(s):  
Difference Sets ◽  
Relative Difference ◽  
Semidirect Products ◽  
Relative Difference Sets ◽  
Amalgamated Subgroup

Another proof of a theorem on difference sets

1967 ◽  
Vol 63 (3) ◽  
pp. 595-596
Author(s):  
D. L. Yates
Keyword(s):  
Cyclic Group ◽  
Elementary Proof ◽  
Existence Theorems ◽  
Difference Sets ◽  
Difference Set ◽  
En Bloc ◽  
Elementary Matrix

Multipliers of a difference set are of great importance in existence theorems, since they enable us to reject many configurations en bloc. (For a description of such theorems, see Mann (1).) The following theorem, which determines those cyclic group difference sets for which −1 is a multiplier, has been proved before by different methods (see, for example, Yamamoto(2) and Johnsen(3); and a more elementary matrix proof by Brualdi(4)) but the following ‘elementary’ proof may be of interest.

The Inverse Multiplier for Abelian Group Difference Sets

1964 ◽  
Vol 16 ◽  
pp. 787-796 ◽  
Author(s):  
E. C. Johnsen
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Collineation Group ◽  
Difference Sets ◽  
Difference Set ◽  
The One ◽  
A Finite Group

In(1)Bruck introduced the notion of a difference set in a finite group. LetGbe a finite group ofvelements and let D = {di},i= 1, . . . ,kbe ak-subset ofGsuch that in the set of differences {di-1dj} each element ≠ 1 inGappears exactly λ times, where 0 < λ <k<v— 1. When this occurs we say that (G,D) is av,k,λ group difference set. Bruck showed that this situation is equivalent to the one where the differences {didj-1} are considered instead, and that av,k, λ group difference set is equivalent to a transitivev,k,λconfiguration, i.e., av,k,λconfiguration which has a collineation group which is transitive and regular on the elements (points) and on the blocks (lines) of the configuration. Among the parametersv,kandλ, then, we have the relation shown by Ryser(5)

scholarly journals Construction of relative difference sets in p -groups

1992 ◽  
Vol 103 (1) ◽  
pp. 7-15 ◽  
Author(s):  
James A. Davis
Keyword(s):  
Difference Sets ◽  
Relative Difference ◽  
Relative Difference Sets
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