A second infinite family of Steiner triple systems without almost parallel classes

For each positive integer n, we construct a Steiner triple system of order v = 2 ( 3 n ) + 1 with no almost parallel class; that is, with no set of v − 1 3 disjoint triples. In fact, we construct families of ( v , k , λ ) -designs with an analogous property. The only previously known examples of Steiner triple systems of order congruent to 1 ( mod 6 ) without almost parallel classes were the projective triple systems of order 2 n − 1 with n odd, and 2 of the 11,084,874,829 Steiner triple systems of order 19.