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(3n)+1 with no almost parallel class; that is, with no set of v−13 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(mod6) without almost parallel classes were the projective triple systems of order 2n−1 with n odd, and 2 of the 11,084,874,829 Steiner triple systems of order 19.