A note on the cyclical generation of disjoint spreads

Abstract

1. It is unknown whether the BIB design PG(ί = 2ra —1, 2): 1 obtained by choosing the points in PG (ί, 2) as treatments and all lines as blocks is resolvable or not for *;>5. C. R. Rao [1], [2] showed that the BIB design PG(ί = 3, 2): 1 with parameters t; = 15, ό = 35, k = 3, r=7, λ = l was resolvable by decomposing all lines in PG(3, 2) into 7 disjoint 1-fold spreads So, Su , S6. The procedure of constructing these spreads is as follows: (1) A set So consisting of 5 lines cyclically generated from the initial line L(x°, x, x) of the minimum cycle 0=5 is chosen as the initial 1-fold spread. (2) Generate Sj+X cyclically by a transformation (T(Sf)= Sj+ι (/=0, 1, ..., 5) where ΰ is a nonsingular linear transformation in PG(3, 2) such that