Uniform Restricted Resolvable Designs with \(r=3\)

Abstract

A Restricted Resolvable Design \(R_rRP(p, k)\) is a resolvable design on \(p\) points with block sizes \(r\) and \(r+1\) in which each point appears \(\alpha\) times. An \(RRP\) is called uniform if all resolution classes consist of the blocks of the same size.
We show that a uniform \(R_3RP(p,\frac{p}{2} -2)\) exists for all \(p \equiv 12 \mod 24, p \neq 12\) except possibly when \(p = 84\) or \(156\).
We also show that if \(g \equiv 3 \mod 6, g \notin \{3, 21, 39\}\) and \(p = 4g \mod 8g\) then there exists an \(R_3RP(p, \frac{p}{2}-(r+1))\) for all

1. \(r \leq \frac{p-4g}{8g}\) if \(\frac{p}{4g}\) is a prime power congruent to \(1 \mod 6\);
2. \(r \leq \frac{p}{4gq}\) where \(q\) is the smallest proper factor of \(\frac{p}{4g}\) if \(\frac{p}{4g}\) is composite and there exists an \(RT(9, \frac{p}{4gp})\).