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New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups

John Polhill
2008 Designs, Codes and Cryptography  
In this paper, we develop three product theorems that construct negative Latin square type partial difference sets and Latin square type partial difference sets in direct products of abelian groups G and  ...  Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups.  ...  Funding was provided by the 2007-08 Bloomsburg University Research and Disciplinary competition.  ... 
doi:10.1007/s10623-007-9165-3 fatcat:7k3tf6uqu5bmhlnltgjbvjfzge

Generalizations of Partial Difference Sets from Cyclotomy to Nonelementary Abelian $p$-Groups

John Polhill
2008 Electronic Journal of Combinatorics  
,r^2+r)$ is called a negative Latin square type partial difference set.  ...  It is believed that this is the first construction of negative Latin square type partial difference sets in nonelementary abelian $p$-groups where the $p$ can be any prime number.  ...  More recently, Polhill constructed negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups [14] .  ... 
doi:10.37236/849 fatcat:bbmez54gp5bcrjmpjxv677avuu

Paley type partial difference sets in non p-groups

John Polhill
2009 Designs, Codes and Cryptography  
By modifying a construction for Hadamard (Menon) difference sets we construct two infinite families of negative Latin square type partial difference sets in groups of the form Z 3 2 × Z p 4t where p is  ...  As a corollary, we are able to construct Paley-Hadamard difference sets of the Stanton-Sprott family in groups of the form Z 3 2 × Z p 4t × Z 9p 4t ±2 when 9p 4t ± 2 is a prime power.  ...  Funding was provided by the 2008-09 Bloomsburg University Research and Disciplinary Competition.  ... 
doi:10.1007/s10623-009-9274-2 fatcat:nwxwniy5ljcizdxwsrxoe3wymu

Partial difference sets and amorphic group schemes from pseudo-quadratic bent functions

Yu Qing Chen, John Polhill
2012 Journal of Algebraic Combinatorics  
We present new abelian partial difference sets and amorphic group schemes of both Latin square type and negative Latin square type in certain abelian p-groups.  ...  Our method is to construct what we call pseudo-quadratic bent functions and use them in place of quadratic forms.  ...  In this paper we construct new negative Latin square type partial difference sets and abelian amorphic group schemes in nonelementary abelian p-groups for all primes p.  ... 
doi:10.1007/s10801-012-0356-2 fatcat:2hkyh5hsazemtii5755pzt25qa

Page 4918 of Mathematical Reviews Vol. , Issue 2003g [page]

2003 Mathematical Reviews  
This paper provides yet another construction of Latin square type PDS in a nonelementary abelian p-group by using fi- nite commutative chain rings. | Qing Xiang (1-DE; Newark, DE) 2003g:05029 05B10 Jia  ...  We show that disjoint (v, 3,1) difference families exist for all v =3 mod 6 with v > 3.” 2003g:05028 05B10 0SBIS 20D15 Hou, Xiang-Dong (1-WRTS; Dayton, OH New partial difference sets in p-groups.  ... 

New constructions of strongly regular Cayley graphs on abelian groups [article]

Koji Momihara
2020 arXiv   pre-print
His result covers all orders of nonelementary abelian groups in which Paley type partial difference sets exist.  ...  proved that there exists a Paley type partial difference set in an abelian group of order 9^iv^4 for any odd positive integer v>1 and any i=0,1.  ...  These parameters are called Latin square type or negative Latin square type depending on whether ǫ = 1 or −1.  ... 
arXiv:2005.03183v3 fatcat:4n63felgwzeg3n6bhrbq27iyeu

Packings of partial difference sets [article]

Jonathan Jedwab, Shuxing Li
2021 arXiv   pre-print
We also study packings of certain negative Latin square type partial difference sets of maximum possible size in abelian groups, all but one of which have identical parameters, and show how to produce  ...  such collections using packings of Latin square type partial difference sets.  ...  We thank the anonymous reviewers for their detailed and helpful comments.  ... 
arXiv:2012.00979v2 fatcat:ommeed6y35ditjgcdtvptu32mm

NEGATIVE LATIN SQUARE TYPE PARTIAL DIFFERENCE SETS IN NONELEMENTARY ABELIAN 2-GROUPS

JAMES A. DAVIS, QING XIANG
2004 Journal of the London Mathematical Society  
Similarly, we construct partial difference sets with Latin square type parameters in the same groups for all k when is even and for all k < when is odd.  ...  square type parameters in nonelementary abelian groups, the groups Z 2k for all k when is odd and for all k < when is even.  ...  Acknowledgement: We thank Frank Fiedler for helping us compute the automorphism groups of some strongly regular graphs. The research of Qing Xiang was supported in part by NSA grant MDA 904-01-1-0036.  ... 
doi:10.1112/s002461070400540x fatcat:zdhxrdsttbgxvh6ogowv4nnfbm

Packings of partial difference sets

Jonathan Jedwab, Shuxing Li
2021 Combinatorial Theory  
We also study packings of certain negative Latin square type partial difference sets of maximum possible size in abelian groups, all but one of which have identical parameters, and show how to produce  ...  such collections using packings of Latin square type partial difference sets.  ...  We thank the anonymous reviewers for their detailed and helpful comments.  ... 
doi:10.5070/c61055384 fatcat:u5s5u75ycjhe7mcwwk7aj5b26u

Negative Latin square type partial difference sets in nonelementary abelian 2-groups [article]

James A. Davis, Qing Xiang
2004 arXiv   pre-print
Similarly, we construct partial difference sets with Latin square type parameters in the same groups for all k when ℓ is even and for all k<ℓ when ℓ is odd.  ...  square type parameters in nonelementary abelian groups, the groups _4^2k×_2^4 ℓ-4k for all k when ℓ is odd and for all k < ℓ when ℓ is even.  ...  Acknowledgement: We thank Frank Fiedler for helping us compute the automorphism groups of some strongly regular graphs.  ... 
arXiv:math/0407423v1 fatcat:o2vvnfb7kjgnjicmm7drtqpgmi

Partial difference sets inp-groups

James A. Davis
1994 Archiv der Mathematik  
D Notice that all of these have the Latin square type as defined in [7]. This gives a new p+1 family of PDS of this type in a nonelementary abelian group. If we set t Corollary 3.1.  ...  This technique has been generalized to aid in the study of other types of difference sets, including PDS and DDS.  ... 
doi:10.1007/bf01189882 fatcat:tqxomp7h4ffupgzd3jjcabbq5e

Rings and constructions of partial difference sets

Xiang-Dong Hou
2003 Discrete Mathematics  
The three constructions provide many new PDS in nonelementary abelian groups.  ...  We present three constructions of partial di erence sets (PDS) using di erent types of ÿnite local rings.  ...  When v = n 2 , a Paley PDS is of Latin square type with r = (n + 1)=2. Latin square type PDS are abundant in elementary abelian groups.  ... 
doi:10.1016/s0012-365x(02)00832-4 fatcat:6gltrotnjbglbaixe3qqs7stmm

The monodromy groups of Schwarzian equations on closed Riemann surfaces [article]

Daniel Gallo, Michael Kapovich, Albert Marden
2000 arXiv   pre-print
_1(R)) be nonelementary.  ...  A branch point is required if and only if the representation \theta does not lift to \SL(2,\C).  ...  Consider two Schottky groups α 1 , α 2 and α 2 , α 3 with a common generator α 2 . Denote the regular sets in S 2 by Ω and Ω ′ , and set R = Ω/ α 1 , α 2 and R ′ = Ω ′ / α 2 , α 3 .  ... 
arXiv:math/9511213v2 fatcat:kc5unplaq5gctjs4vlfjgi42le

The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces

Daniel Gallo, Michael Kapovich, Albert Marden
2000 Annals of Mathematics  
Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that θ(π 1 (R)) be nonelementary  ...  A branch point is required if and only if the representation θ does not lift to SL(2, C). Contents 1. Introduction and background 2. Fixed points of Möbius transformations A.  ...  Consider two Schottky groups α 1 , α 2 and α 2 , α 3 with a common generator α 2 . Denote the regular sets in S 2 by Ω and Ω , and set R = Ω/ α 1 , α 2 and R = Ω / α 2 , α 3 .  ... 
doi:10.2307/121044 fatcat:opgs6techvcefpvvvyjgovkedy

Nonstandard Finite Difference Schemes

Ronald E. Mickens
2018 Notices of the American Mathematical Society  
Dedè and C. Vergara from Politecnico di Milano, and A. Manzoni from EPFL are gratefully acknowledged. Mathematical Modeling, Analysis and Applications in  ...  The first was a model for a data set of type 2 diabetics constructed by a group at the Mt. Sinai School of Medicine.  ...  In addition to mapping class groups, examples of groups admitting nonelementary acylindrical actions on hyperbolic spaces include nonelementary hyperbolic groups, groups which are nonelementary hyperbolic  ... 
doi:10.1090/noti1617 fatcat:nzhfdj7fsncndnhfiaegkizvue
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