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We prove several structural properties of Steiner triple systems (STS) oforder 3w+3 that include one or more transversal subdesigns TD(3,w). Using anexhaustive search, we find that there are 2004720 isomorphism classes ofSTS(21) including a subdesign TD(3,6), or, equivalently, a 6-by-6 latin square.

The Steiner triple systems of order 21 with a transversal subdesign TD(3,6)