ABSTRACT
Apart from the infinite classes, there are a number of conical flocks of small order. Actually, all flocks of prime-power order ≤ 29 are known, mostly by the use of the computer, and the flocks of order 32 are completely known. The use of the computer is probably pushed about as far as it can be, at least with current memory and speed. For example, it is estimated by Law and Penttila [913] that it might require eight years of computer time to determine the flocks of order 31. Still it is important to know flocks of small order, as these conceivably point to infinite classes. For example, there is a flock of order 11 (i.e., in PG(3, 11)) whose associated translation plane q2 admits a collineation group of order q(q +1) in the translation complement. Jha and Johnson [632, 633] completely determine the translation planes of order q2 with spreads in PG(3, q) that admit linear groups of order q(q+1). When q is odd, all of these planes are either conical flock planes or Ostrom derivates of conical flock planes. The classification method is one of structure, in that in all but two sporadic cases, such planes are shown to be related to a Desarguesian plane by a single or multiple nest replacement procedure. For example, the Fisher conical flock planes are constructed from a Desarguesian affine plane by q-nest replacement. The more general concept of multiple nest replacements is developed in Jha and Johnson [632, 633] to analyze the planes obtained. In particular, the De Clerck-Herssens-Thas conical flock plane of order 11, originally found by computer, admits a collineation group of order q(q + 1), when q = 11, and may be shown to be constructed from a Desarguesian plane by double-nest replacement.
