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Negative Latin Square type Partial Difference Sets in Nonelementary Abelian 2‐Groups
Author(s) -
Davis James A.,
Xiang Qing
Publication year - 2004
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070400540x
Subject(s) - mathematics , bijection , abelian group , combinatorics , type (biology) , latin square , strongly regular graph , unit (ring theory) , finite field , algebraic number , discrete mathematics , mathematical analysis , pathwidth , ecology , graph , rumen , chemistry , mathematics education , food science , line graph , fermentation , biology
Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, the first known family of partial difference sets with negative Latin square type parameters is constructed in nonelementary abelian groups, the groupsZ 4 2 k×Z 4 4 l − 4 kfor all k when ℓ is odd and for all k < ℓ when ℓ is even. Similarly, partial difference sets with Latin square type parameters are constructed in the same groups for all k when ℓ is even and for all k <ℓ when ℓ is odd. These constructions provide the first example where the non‐homomorphic bijection approach outlined by Hagita and Schmidt can produce difference sets in groups that previously had no known constructions. Computer computations indicate that the strongly regular graphs associated to the partial difference sets are not isomorphic to the known graphs, and it is conjectured that the family of strongly regular graphs will be new.