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

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. ...

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

2008
*
Electronic Journal of Combinatorics
*

,r^

doi:10.37236/849
fatcat:bbmez54gp5bcrjmpjxv677avuu
*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] . ...##
###
Paley type partial difference sets in non p-groups

2009
*
Designs, Codes and Cryptography
*

By modifying a construction for Hadamard (Menon)

doi:10.1007/s10623-009-9274-2
fatcat:nwxwniy5ljcizdxwsrxoe3wymu
*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. ...##
###
Partial difference sets and amorphic group schemes from pseudo-quadratic bent functions

2012
*
Journal of Algebraic Combinatorics
*

We present

doi:10.1007/s10801-012-0356-2
fatcat:2hkyh5hsazemtii5755pzt25qa
*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. ...##
###
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]

2020
*
arXiv
*
pre-print

His result covers all orders of

arXiv:2005.03183v3
fatcat:4n63felgwzeg3n6bhrbq27iyeu
*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. ...##
###
Packings of partial difference sets
[article]

2021
*
arXiv
*
pre-print

We also study packings of certain

arXiv:2012.00979v2
fatcat:ommeed6y35ditjgcdtvptu32mm
*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. ...##
###
NEGATIVE LATIN SQUARE TYPE PARTIAL DIFFERENCE SETS IN NONELEMENTARY ABELIAN 2-GROUPS

2004
*
Journal of the London Mathematical Society
*

Similarly, we construct

doi:10.1112/s002461070400540x
fatcat:zdhxrdsttbgxvh6ogowv4nnfbm
*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. ...##
###
Packings of partial difference sets

2021
*
Combinatorial Theory
*

We also study packings of certain

doi:10.5070/c61055384
fatcat:u5s5u75ycjhe7mcwwk7aj5b26u
*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. ...##
###
Negative Latin square type partial difference sets in nonelementary abelian 2-groups
[article]

2004
*
arXiv
*
pre-print

Similarly, we construct

arXiv:math/0407423v1
fatcat:o2vvnfb7kjgnjicmm7drtqpgmi
*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. ...##
###
Partial difference sets inp-groups

1994
*
Archiv der Mathematik
*

D Notice that all of these have the

doi:10.1007/bf01189882
fatcat:tqxomp7h4ffupgzd3jjcabbq5e
*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. ...##
###
Rings and constructions of partial difference sets

2003
*
Discrete Mathematics
*

The three constructions provide many

doi:10.1016/s0012-365x(02)00832-4
fatcat:6gltrotnjbglbaixe3qqs7stmm
*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*. ...##
###
The monodromy groups of Schwarzian equations on closed Riemann surfaces
[article]

2000
*
arXiv
*
pre-print

_1(R)) be

arXiv:math/9511213v2
fatcat:kc5unplaq5gctjs4vlfjgi42le
*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*. ...##
###
The Monodromy Groups of Schwarzian Equations on Closed Riemann Surfaces

2000
*
Annals of Mathematics
*

Necessary

doi:10.2307/121044
fatcat:opgs6techvcefpvvvyjgovkedy
*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*. ...##
###
Nonstandard Finite Difference Schemes

2018
*
Notices of the American Mathematical Society
*

Dedè

doi:10.1090/noti1617
fatcat:nzhfdj7fsncndnhfiaegkizvue
*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 ...
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