Uniformly resolvable designs with index one, block sizes three and five and up to five parallel classes with blocks of size five

Each parallel class of a uniformly resolvable design (URD) contains blocks of only one block size k (denoted k-pc). The number of k-pcs is denoted rk. The necessary conditions for URDs with v points, index one, blocks of size 3 and 5, and r3, r5 > 0, are v ≡ 15 (mod 30). If rk > 1, then v ≥ k2, and r3 = (v−1−4 · r5)/2. For r5 = 1 these URDs are known as group divisible designs. We prove that these necessary conditions are sufficient for r5 = 3 except possibly v = 105, and for r5 = 2, 4, 5 with possible exceptions (v = 105, 165, 285, 345) New labeled frames and labeled URDs, which give new URDs as ingredient designs for recursive constructions, are the key in the proofs. © 2009 Elsevier B.V. All rights reserved.