The existence of some resolvable block designs with divisibility into groups

B. T. Rumov

Matematičeskie zametki, Tome 19 (1976) no. 4, pp. 623-634.

Résumé

This paper proves the existence of resolvable block designs with divisibility into groups $GD(v;k,m;\lambda_1,\lambda_2)$ without repeated blocks and with arbitrary parameters such that $\lambda_1=k$, $(v-1)/(k-1)\le\lambda_2\le v^{k-2}$ (and also $\lambda_1\le k/2$), $(v-1)/(2(k-1))\le\lambda_2\le v^{k-2}$ in case $k$ is even) $k\ge4$ and $p\equiv1\pmod{k-1}$, $k$ for each prime divisor $p$ of number $v$. As a corollary, the existence of a resolvable $BIB$-design $(v,k,\lambda)$ without repeated blocks is deduced with $\lambda=k$ (and also with $\lambda=k/2$ in case of even $k$) $k>\sqrt{p}v=pk^\alpha$ , where $\alpha$ is a natural number if $k$ is a prime power $\alpha=1$ if $k$ is a composite number.