A Steiner quadruple system is an ordered pair where V is a finite set of symbols and B is a set of 4-element subsets of V called quadruples with the property that every 3-element subset of V is a subset of exactly one quadruple in B. |V| is called the order of the Steiner quadruple system. For short, we write SQS(v) to denote a Steiner quadruple system of order v.
It is very important to remember that two quadruples in a Steiner quadruple system can intersect in two symbols, since it is only the 3-element subsets that are required to occur in a unique quadruple. (For those of us who are most familiar with block designs, this is easy to forget!)