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Small Uniformly Resolvable Designs for Block Sizes 3 and 4
Author(s) -
Schuster Ernst
Publication year - 2013
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.21361
Subject(s) - mathematics , mod , combinatorics , block (permutation group theory) , block size , class (philosophy) , discrete mathematics , computer science , artificial intelligence , key (lock) , computer security
A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k , such a class is denoted k ‐pc and for a given k the number of k ‐pcs is denoted r k . In this paper, we consider the case of block sizes 3 and 4 (both existent). We use v to denote the number of points, in this case the necessary conditions imply that v ≡ 0 (mod 12). We prove that all admissible URDs with v < 200 points exist, with the possible exceptions of 13 values of r 4 over all permissible v . We obtain a URD({3, 4}; 276) with r 4 = 9 by direct construction use it to and complete the construction of all URD({3, 4}; v ) with r 4 = 9. We prove that all admissible URDs for v ≡ 36 (mod 144), v ≡ 0 (mod 60), v ≡ 36 (mod 108), and v ≡ 24 (mod 48) exist, with a few possible exceptions. Recently, the existence of URDs for all admissible parameter sets with v ≡ 0 (mod 48) was settled, this together with the latter result gives the existence all admissible URDs for v ≡ 0 (mod 24), with a few possible exceptions.