On Point-Cyclic Resolutions of the 2-(63,7,15) Design Associated with PG(5,2)

摘要

A t(v,k,u) design is a set of v points together with a collection of its k-subsets called blocks so that t points are contained in exactly u blocks. PG(n,q), the n-dimensional projective geometry over GF(q) is a 2(qn+qnm1+>+q+1,q2+q+1, qnm2+ qnm3+>+q+1) design when we take its points as the points of the design and its planes as the blocks of the design. A 2(v,k,u) design is said to be resolvable if the blocks can be partitioned as ?={R1,R2,…,Rs}, where s=u(vm1)/(km1) and each Ri consists of v/k disjoint blocks. If a resolvable design has an automorphism ? which acts as a cycle of length v on the points and ??=?, then the design is said to be point-cyclically resolvable. The design consisting of points and planes of PG(5,2) is shown to be point-cyclically resolvable by enumerating all inequivalent resolutions which are invariant under a cyclic automorphism group G=d?￠ where ? is a cycle of length v. These resolutions are shown to be the only resolutions which admit point-transitive automorphism group.