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 kk. Such a class is denoted kk-pc and for a given kk the number of kk-pcs is denoted rkrk. Let vv 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(mod12). It has been shown that almost all URDs with permissible r3r3 and r4r4 exist for v≡0(mod24), v≡0(mod60), v≡36(mod144) or v≡36(mod108). 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=12v=12, r3=1r3=1 and r4=3r4=3.