The existence of some resolvable block designs with divisibility into groups

Abstract

This paper proves the existence of resolvable block designs with divisibility into groups GD(v; k, m; λ₁, λ₂) without repeated blocks and with arbitrary parameters such that λ₁ = k, (v−1)/(k−1) ≤ λ₂ ≤ v^k−2 (and also λ₁ ≤ k/2, (v−1)/(2(k−1)) ≤ λ₂ ≤ v^k−2 in case k is even) k ≥ 4 andp=1 (mod k−1), k < p for each prime divisor p of number v. As a corollary, the existence of a resolvable BIB-design (v, k, λ) without repeated blocks is deduced with X = k (and also with λ = k/2 in case of even k) k ⋗$\sqrt p v = pk^\alpha$, where a is a natural number if k is a prime power andα=1 if k is a composite number.