Computational and Constructive Design Theory pp 255-335 | Cite as

# Another Look at Large Sets of Steiner Triple Systems

Chapter

## Abstract

If *v* = 1 or 3 mod 6 and *v* > 7, then there exists a large set of Steiner triple systems of order *v*, *LS* (*STS* (*v*)).

This result was largely known by 1984, though the six cases which were unsettled then have since been solved, and the proof simplified. The proof is by construction, using a fairly small number of initial large sets. Each construction is explained, often with the aid of tables of blocks. In the interests of brevity, proofs and complete examples are not given, though the ingredients needed for the construction of the examples are. The recursive constructions are of three types:

- (i)
extension of a large set of

*STS*(*v*) to a large set of*STS*(3*v*) by using idempotent commutative quasigroups; - (ii)
extension of a large set of

*STS*(*v*)to a large set of*STS*(2*v*+ 1) by using good one-factorizations; - (iii)
special constructions to fill in the remaining cases.

## Keywords

Triple System Counting Argument Steiner Triple System Balance Incomplete Block Design Transversal Design
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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