The first infinite family of orthogonal Steiner systems S(3,5,v)

Abstract

The research on orthogonal Steiner systems S(t,k,v) was initiated in 1968. For (t,k)∈{(2,3),(3,4)}, this corresponds to orthogonal Steiner triple systems (STSs) and Steiner quadruple systems (SQSs), respectively. The existence problem of a pair of orthogonal STSs or SQSs was settled completely thirty years ago. However, for Steiner systems with t≥3 and k≥5, only two small examples of orthogonal pairs were known to exist before this work. An infinite family of orthogonal Steiner systems S(3,5,v) is constructed in this paper. In particular, the existence of a pair of orthogonal Steiner systems S(3,5,4m+1) is established for any even m≥2; additionally a pair of orthogonal G-designs G(4m+15,5,5,3) is displayed for any odd m≥3. The construction is based on the Steiner systems admitting 3-transitive automorphism groups supported by elementary symmetric polynomials. Moreover, 50 mutually orthogonal Steiner systems S(5,8,24) are shown to exist.