Non-existence results for pairwise balanced designs with block sizes 4 and k


This table extends the known non-existence results in Table 3.17 (page 206) in the CRC Handbook of Combinatorial Designs, when the block size set is K={4,k} with 9< k<=15. I do hope that these results are useful for somebody who tries to investigate the existence question.
Notation: N is used when the closure (the set of integers v such that there exists a PBD[v,K]) contains all sufficiently large integers and x1,x2,...,xn mod y denotes the set of all nonnegative integers which are congruent x1 or x2 or ... or xn modulo y.
Remark: This table contains only NON-EXISTENCE results. For existence results, open cases with k<=9 and references see for example Mullin,Ling,Abel and Bennett: On the Closure of Subsets of {4,5,...,9} which contain 4. ARS Combinatoria 45 (1997), pp. 33-76.

Subset Closure Genuine Exceptions
{4} 1 4 mod 12 -
{4 5} 0 1 mod 4 8 9 12
{4 6} 0 1 mod 3 7 9 10 12 15 18 19 22 24 27
{4 7} 1 mod 3 10 19
{4 8} 0 1 mod 4 5 9 12 17 20 21 24 33 41 44 45
{4 9} 0 1 4 9 mod 12 12 21 24 48
{4 10} 1 mod 3 7 19 22
{4 11} N 5 6 7 8 9 10 12 14 15 17 18 19 20 21 22 23 24 26 27 29 30 31 32 33 34 35 36 38 39 42 43 45 46 47 48 50 51 53 54 55 56 57 58 59 60 62 63 65 67 68 69 70 72 75 77 79 80 81 82 83 86 89 92 95 98
{4 12} 0 1 4 9 mod 12 9 21 24 33 36 57 69
{4 14} N 5 6 7 8 9 10 11 12 15 17 18 19 20 21 22 23 24 26 27 29 30 31 32 33 34 35 36 38 39 41 42 43 44 45 46 47 48 50 51 54 55 57 58 59 60 62 63 65 66 67 68 69 70 71 72 74 75 77 78 79 80 81 82 83 84 86 87 89 90 91 92 93 94 96 99 101 102 103 104 105 106 107 108 110 111 113 114 115 116 117 118 119 120 122 127 130 131 134 139 149 152
{4 15} 0 1 mod 3 6 7 9 10 12 18 19 21 22 24 27 30 31 33 34 36 39 42 43 45 46 48 51 54 55 58 63 66 67 69 70 72 78 79 81 82 84 87 90 91 93 94 96 103 105 106 108 111 114 115 118 123 127 129 130 132 138 139 142 151 154

This research was done while the author visited the Faculty of Mathematics, Department of Combinatorics and Optimization at the University of Waterloo, Canada; he thanks the Department for its hospitality. The non-existence of a PBD[69,{4.12}] was proved in joint work with Rolf Rees.
Martin Grüttmüller, 09.11.98
mgruttm@luna.math.uni-rostock.de