Quasi-symmetric Designs

The importance of coding theory as a valuable tool in the study of designs has been known for quite some time. We mention, for example, M. Hall, Jr., MacWilliams and Sloane, Pless, and also the monographs by Cameron and van Lint and Tonchev. Recently Tonchev, Calderbank, and Bagchi have proved some very nice results about designs using coding theory. We have referred to Bagchi's result (Theorem 7.30) in an earlier chapter.

The paper of Tonchev has shown the link between quasi-symmetric designs and self-dual codes. Calderbank, has proved some elegant non-existence criteria about 2-designs in terms of their intersection numbers. The proof of one of Calderbank's results depends on some deep theorems of Gleason and Mallows, and MacWilliams-Sloane on weight enumerators of certain self-dual codes. The results of Calderbank, and Tonchev when specialized to quasi-symmetric designs give strong results about existence, non-existence or uniqueness. For example, Tonchev shows the falsity of a part of the well known Hamada conjecture concerning the rank of the incidence matrix of certain 2-designs. Some results of Tonchev and Calderbank, seem to have been motivated by Neumaier's table of exceptional quasi-symmetric designs given in Chapter VIII.

The purpose of this chapter is to review some of the results of Tonchev, and Calderbank, which rely on codes as one of their principal tools.