A Few More Partitioned Balanced Tournament Designs

A balanced tournament design, \(\mathrm{BTD}(n)\), defined on a \(2n\)-set \(V\), is an arrangement of the \(\binom{2n}{2}\) distinct unordered pairs of the elements of \(V\) into an \(n \times 2n-1\) array such that:
(1) every element of \(V\) is contained in precisely one cell of each column, and
(2) every element of \(V\) is contained in at most two cells of each row.

If we can partition the columns of a \(\mathrm{BTD}(n)\) defined on \(V\) into three sets \(C_1, C_2, C_3\) of sizes \(1, n-1, n-1\) respectively such that the columns in \(C_1 \cup C_2\) form a Howell design of side \(m\) and order \(2n\), an \(\mathrm{H}(n,2n)\), and the columns in \(C_1 \cup C_3\) form an \(\mathrm{H}(n,2n)\), then the \(\mathrm{BTD}(n)\) is called partitionable. We denote a partitioned balanced tournament design of side \(n\) by \(\mathrm{PBTD}(n)\). The existence of these designs has been determined except for seven possible exceptions. In this note, we describe constructions for four of these designs. This completes the spectrum of \(\mathrm{PBTD}(n)\) for \(n\) even.