Uniformly resolvable designs with block sizes 3 and 4

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 . Let v denote the number of points of the URD. For the case of block sizes 3 and 4 (both existing), the necessary conditions imply that v ? 0 ( mod 12 ) . It has been shown that almost all URDs with permissible r 3 and r 4 exist for v ? 0 ( mod 24 ) , v ? 0 ( mod 60 ) , v ? 36 ( mod 144 ) or v ? 36 ( mod 108 ) . In this paper, we prove that the necessary conditions for the existence of a URD with block sizes 3 and 4 are also sufficient, except when v = 12 , r 3 = 1 and r 4 = 3 .

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