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On simple radical difference families
Author(s) -
Buratti Marco
Publication year - 1995
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.3180030208
Subject(s) - mathematics , coset , combinatorics , multiplicative function , root of unity , prime power , prime (order theory) , simple group , group (periodic table) , base (topology) , simple (philosophy) , discrete mathematics , mathematical analysis , philosophy , chemistry , physics , organic chemistry , epistemology , quantum mechanics , quantum
Abstract For q a prime power and k odd (even), we define a ( q,k ,1) difference family to be radical if each base block is a coset of the k th roots of unity in the multiplicative group of GF( q ) (the union of a coset of the ( k − 1)th roots of unity in the multiplicative group of GF( q ) with zero). Such a family will be denoted by RDF. The main result on this subject is a theorem dated 1972 by R.M. Wilson; it is a sufficient condition for the existence of a ( q,k , 1)‐RDF for any k . We improve this result by replacing Wilson's condition with another sufficient but weaker condition, which is proved to be necessary at least for k ⩽ 7. As a consequence, we get new difference families and hence new Steiner 2‐designs. © 1995 John Wiley & Sons, Inc.