Symmetric balanced ternary designs with \(\rho_ 1=1\) or \(2\) (Q1323540)

A balanced ternary design is a collection of blocks containing various points 0, 1 or 2 times and such that the size of the blocks and the number of occurrences of any point or any pair of distinct points simultaneously in the same block are constants. The parameters are written in the form \((V,B,\rho_ 1,\rho_ 2,R,K,\Lambda)\), where \(V\) is the number of points, \(B\) is the number of blocks, \(\rho_ i\) is the number of blocks containing a given point \(i\) times for \(i=1\) or 2, \(R\) is a number of times a given point occurs in a block, \(K\) is the size of any block and \(\Lambda\) is the number of simultaneous occurrences of an arbitrary pair of distinct points in a block. Properties of BTDs can be found in \textit{E. J. Billington} and \textit{P. J. Robinson} [Ars Comb. 16, 235-258 (1983; Zbl 0534.05010)] and \textit{E. J. Billington} [Ars Comb. 17- A, 37-72 (1984; Zbl 0537.05004)]. The main results of this paper are formulated in the following two non-existence theorems for symmetric balanced ternary designs: If \(\rho_ 1=1\) and \(\Lambda\equiv 0\pmod 4\), then either \(V=\Lambda+1\) or \(4\rho_ 2-\Lambda+1\) is a square and \(\sqrt{4\rho_ 2-\Lambda+1}\) divides \(\Lambda^ 2-1\). If \(\rho_ 2=2\), then \(V=((m+1)/2)^ 2+2\), \(K=(m^ 2+7)/4\) and \(\Lambda=((m-1)/2)^ 2+1\), where \(m\equiv 3\pmod 4\). An example belonging to the latter series with \(V=18\) is constructed.