Designs from pairs of finite fields. A cyclic unital U(6) and other regular steiner 2-designs

Abstract

A general construction for Steiner 2-designs with prime power block size (and with a point-regular automorphism group) is presented. Its success depends on number-theoretic restrictions on the parameters-these are completely analysed in case of block sizes k ≤11. The new designs constructed include infinitely many cyclic Steiner 2-designs with block size 7. Among them is a cyclic unital U(6), that is, an S(2, 6 + 1, 6³ + 1). It is the first example of a unital with non-prime power parameter and the second example of a cyclic unital.