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.

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 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 A family of skew Hadamard difference sets

2006 ◽  
Vol 113 (7) ◽  
pp. 1526-1535 ◽  
Author(s):  
Cunsheng Ding ◽  
Jin Yuan
Keyword(s):  
Difference Sets ◽  
Skew Hadamard Difference Sets ◽  
Hadamard Difference Sets

Some results on skew Hadamard difference sets

2008 ◽  
Vol 50 (1) ◽  
pp. 93-105 ◽  
Author(s):  
Guobiao Weng ◽  
Lei Hu
Keyword(s):  
Difference Sets ◽  
Skew Hadamard Difference Sets ◽  
Hadamard Difference Sets

scholarly journals Cyclotomic constructions of skew Hadamard difference sets

2012 ◽  
Vol 119 (1) ◽  
pp. 245-256 ◽  
Author(s):  
Tao Feng ◽  
Qing Xiang
Keyword(s):  
Difference Sets ◽  
Skew Hadamard Difference Sets ◽  
Hadamard Difference Sets

scholarly journals Skew Hadamard difference sets from the Ree–Tits slice symplectic spreads in PG(3,32h+1)

2007 ◽  
Vol 114 (5) ◽  
pp. 867-887 ◽  
Author(s):  
Cunsheng Ding ◽  
Zeying Wang ◽  
Qing Xiang
Keyword(s):  
Difference Sets ◽  
Skew Hadamard Difference Sets ◽  
Symplectic Spreads ◽  
Hadamard Difference Sets

scholarly journals Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes

COMBINATORICA ◽  
2014 ◽  
Vol 35 (4) ◽  
pp. 413-434 ◽  
Author(s):  
Tao Feng ◽  
Koji Momihara ◽  
Qing Xiang
Keyword(s):  
Cayley Graphs ◽  
Difference Sets ◽  
Skew Hadamard Difference Sets ◽  
Strongly Regular ◽  
Hadamard Difference Sets
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