Class-uniformly resolvable designs with all but one block having size two

A Class-Uniformly Resolvable Design (CURD) is a resolvable design in which each parallel class has the same block structure.

We study CURDS in which each parallel class contains one block of size $m$ and the remaining blocks have size $2$, for $m \ge 3$.

In addition to establishing necessary conditions for such a CURD to exist, we present two general constructions.

The first transforms a particular type of cyclic design with block size $k$ into a CURD with partition $m^12^{\frac{n-m}{2}}$ where $m = 2k$.

This construction is used to generate CURDS with 26 varieties (where $m=6$) and with 82 varieties (where $m=10$).

The second constructs a CURD with partition $m^12^{\frac{n-m}{2}}$ for every value of $m$ that is the power of an odd prime.