More \(c\)-Bhaskar Rao designs with small block size
A \(c\)-Bhaskar Rao design, a \(c\)-\(\text{BRD}(v,k,\lambda)\) is identified with a \(\{-1,0,1\}\)-matrix whose rows have inner product \(c\). When the \(-1\)'s are changed to \(+1\), the matrix is the incidence matrix of a \(\text{BIBD}(v,k,\lambda)\). When \(c=0\), the designs have been called BRDs. In a previous article [\textit{M. Greig}, \textit{S. P. Hurd}, \textit{J. S. McCranie} and \textit{D. G. Sarvate}, On \(c\)-Bhaskar Rao designs with block size 4, J. Comb. Des. 10, 361-386 (2002)] the spectrum of \(c\)-BRDs was studied for block size 4. The present paper complements the other in that most of the designs not shown in the first are exhibited here. In particular, all remaining cases for \(c\) not zero are given. The techniques are varied and interesting and include using group design BRDs. For \(c=0\), the classical BRD case, much progress is made on the (only remaining) case with \(v\equiv 4\pmod 6\) and \(\lambda= 2\). There are at most 30 smaller exceptions. In the former paper it was shown that a \(\text{BIBD}(v,k,k-1)\) can only be signed into a \((-1)\)-\(\text{BRD}(v,k,k-1)\) when \(k\) is even and every block receives \(k/2\) minus signs. Consequently using the signs in a BRD to divide the block into subblocks of size \(k/2\), we see these subblocks must form a \(\text{BIBD}(v,k/2,k/2-1)\). Thus a \((-1)\)-\(\text{BRD}(v,k,k-1)\) gives an example of a \(\text{BIBD}(v,k,k-1)\) with a nested \(\text{BIBD}(v,k/2,k/2- 1)\). The existence of \((-1)\)-\(\text{BRD}(16,10,9)\) is of particular interest as it corresponds to the only open case [\textit{J. P. Morgan}, \textit{D. A. Preece} and \textit{D. H. Rees}, Discrete Math. 231, 351-389 (2001; Zbl 0988.05018)] and is herein resolved.
scientific article; zbMATH DE number 1882218