Simple 8-Designs with Small Parameters

Authors:
Anton Betten;Adalbert Kerber;Reinhard Laue;Alfred Wassermann
Affiliations:
Mathematical Department, University of Bayreuth, D-95440 Bayreuth;Mathematical Department, University of Bayreuth, D-95440 Bayreuth;Mathematical Department, University of Bayreuth, D-95440 Bayreuth;Mathematical Department, University of Bayreuth, D-95440 Bayreuth
Venue:
Designs, Codes and Cryptography
Year:
1998

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Abstract

We show the existence of simple 8-(31,10,93) and 8-(31,10,100)designs. For each value of λ we show 3 designs in fulldetail. The designs are constructed with a prescribed group of automorphismsPSL(3,5) using the method of Kramer and Mesner KramerMesner76.They are the first 8-designs with small parameters which are knownexplicitly. We do not yet know if PSL(3,5) is the full groupof automorphisms of the given designs. There are altogether 138 designs withλ = 93 and 1658 designs with λ = 100 andPSL(3,5) as a group of automorphisms. We prove that they areall pairwise non-isomorphic. For this purpose, a brief account on theintersection numbers of these designs is given. The proof is done in twodifferent ways. At first, a quite general group theoretic observation showsthat there are no isomorphisms. In a second approach we use the blockintersection types as invariants, they classify the designs completely.