The Steiner Quadruple Systems of order 16, SQS(16), were classified by Petteri Kaski, Patric Östergård and Olli Pottonen in 2005; there are 1,054,163 such systems. The systems, and also all smaller Steiner Quadruple Systems are available on this page.
The data format is as follows. The files are in ASCII, one system on each line. The point set is {a, b, c, … p} (for v=16; for smaller v an obvious subset). The first four characters are the first block, next four characters the second block etc. The largest file is compressed.