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New necessary conditions on Paley‐type partial difference sets in abelian groups
Author(s) -
Wang Zeying
Publication year - 2019
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21655
Subject(s) - mathematics , abelian group , modulo , prime (order theory) , order (exchange) , type (biology) , combinatorics , group (periodic table) , hadamard transform , discrete mathematics , pure mathematics , mathematical analysis , ecology , biology , chemistry , organic chemistry , finance , economics
Let G be an abelian group of order v , where v = p 1 2 k 1p 2 2 k 2⋯ p n 2 k n, n ≥ 2 , p 1 , p 2 , … , p n are distinct odd prime numbers. In this paper, we prove that if G contains a regular Paley‐type partial difference set (PDS), then for any 1 ≤ i ≤ n , p i is congruent to 3 modulo 4 whenever k i is odd. These new necessary conditions further limit the specific order of an abelian group G in which there can exist a Paley‐type PDS. Our result is similar to a result on abelian Hadamard (Menon) difference sets proved by Ray‐Chaudhuri and Xiang in 1997.