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Old and new designs via difference multisets and strong difference families
Author(s) -
Buratti Marco
Publication year - 1999
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/(sici)1520-6610(1999)7:6<406::aid-jcd2>3.0.co;2-u
Subject(s) - multiset , mathematics , combinatorics , series (stratigraphy) , block (permutation group theory) , block design , group (periodic table) , discrete mathematics , physics , paleontology , quantum mechanics , biology
Let X =(( x 1,1 , x 1,2 ,…, x 1, k ),( x 2,1 , x 2,2 ,…, x 2, k ),…,( x t ,1 , x t ,2 ,…, x t,k )) be a family of t multisets of size k defined on an additive group G . We say that X is a t ‐( G,k, μ) strong difference family (SDF) if the list of differences ( x h,i ‐x h,j ∣ h =1,…, t;i ≠ j ) covers all of G exactly μ times. If a SDF consists of a single multiset X , we simply say that X is a ( G,k, μ) difference multiset. After giving some constructions for SDF's, we show that they allow us to obtain a very useful method for constructing regular group divisible designs and regular (or 1‐rotational) balanced incomplete block designs. In particular cases this construction method has been implicitly used by many authors, but strangely, a systematic treatment seems to be lacking. Among the main consequences of our research, we find new series of regular BIBD's and new series of 1‐rotational (in many cases resovable) BIBD's.