Abstract
Packings (resolutions) of designs have been of interest to combinatorialists in recent years as a way of creating new designs from old ones. Line packings of projective 3-space were the first packings studied, but it is still unknown when a partial packing can be completed to a packing in this case. In this paper we show that there is no guarantee from a combinatorial point of view of completing such a partial packing even when the deficiency is 2. In particular, we construct for every odd prime power q a set of 2(q 2+1) lines which doubly cover the points of PG(3,q) and yet cannot be partitioned into two spreads (resolution classes). The method is based on manipulations of primitive elements of finite fields.
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References
Bruck, R. H., ‘Construction Problems of Finite Projective Planes’ in Combinatorial Mathematics and its Applications (R. C. Bose and T. A. Dowling (eds)), University of North Carolina Press, Chapel Hill, 1969, pp. 426–514.
Denniston, R. H. F., ‘Packings of PG(3,q)’, in Finite Geometric Structures and their Applications (A. Barlotti (ed.)), Edizioni Cremonese, Rome, 1973, pp. 193–199.
Ebert, G. L., ‘Partitioning Projective Geometries into Caps’ (to appear in Canad. J. Math.).
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This work was supported in part by the National Research Council of Canada.
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Ebert, G.L. The completion problem for partial packings. Geom Dedicata 18, 261–267 (1985). https://doi.org/10.1007/BF00181225
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DOI: https://doi.org/10.1007/BF00181225