scholarly journals A Recursive Construction for Skew Hadamard Difference Sets

10.37236/9058 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Koji Momihara
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Difference Sets ◽  
Difference Set ◽  
Recursive Construction ◽  
Skew Hadamard Difference Sets ◽  
New Construction ◽  
Hadamard Difference Sets ◽  
Skew Hadamard Difference Set ◽  
Hadamard Difference Set

A major conjecture on the existence of abelian skew Hadamard difference sets is: if an abelian group $G$ contains a skew Hadamard difference set, then $G$ must be elementary abelian. This conjecture remains open in general. In this paper, we give a recursive construction for skew Hadamard difference sets in abelian (not necessarily elementary abelian) groups. The new construction can be considered as a result on the aforementioned conjecture: if there exists a counterexample to the conjecture, then there exist infinitely many counterexamples to it.

scholarly journals Inequivalence of Skew Hadamard Difference Sets and Triple Intersection Numbers Modulo a Prime

10.37236/3762 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Koji Momihara
Keyword(s):  
Abelian Groups ◽  
Difference Sets ◽  
Intersection Numbers ◽  
Skew Hadamard Difference Sets ◽  
Triple Intersection ◽  
New Construction ◽  
Hadamard Difference Sets

Recently, Feng and Xiang found a new construction of skew Hadamard difference sets in elementary abelian groups. In this paper, we introduce a new invariant for equivalence of skew Hadamard difference sets, namely triple intersection numbers modulo a prime, and discuss inequivalence between Feng-Xiang skew Hadamard difference sets and the Paley difference sets. As a consequence, we show that their construction produces infinitely many skew Hadamard difference sets inequivalent to the Paley difference sets.

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 Tiling Groups with Difference Sets

10.37236/5157 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Ante Ćustić ◽  
Vedran Krčadinac ◽  
Yue Zhou
Keyword(s):  
Difference Sets ◽  
Skew Hadamard Difference Sets ◽  
Existence Conditions ◽  
Hadamard Difference Sets ◽  
Necessary Existence

We study tilings of groups with mutually disjoint difference sets. Some necessary existence conditions are proved and shown not to be sufficient. In the case of tilings with two difference sets we show the equivalence to skew Hadamard difference sets, and prove that they must be normalized if the group is abelian. Furthermore, we present some constructions of tilings based on cyclotomy and investigate tilings consisting of Singer difference sets.

scholarly journals Sets with Few Differences in Abelian Groups

10.37236/5502 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Mitchell Lee
Keyword(s):  
Abelian Group ◽  
Explicit Formula ◽  
Abelian Groups ◽  
Difference Set ◽  
The Difference ◽  
Fixed Cardinality

Let $(G, +)$ be an abelian group. In 2004, Eliahou and Kervaire found an explicit formula for the smallest possible cardinality of the sumset $A+A$, where $A \subseteq G$ has fixed cardinality $r$. We consider instead the smallest possible cardinality of the difference set $A-A$, which is always greater than or equal to the smallest possible cardinality of $A+A$ and can be strictly greater. We conjecture a formula for this quantity and prove the conjecture in the case that $G$ is an elementary abelian $p$-group. This resolves a conjecture of Bajnok and Matzke on signed sumsets.

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