On Point-cyclic Parallelisms of Pg(3, 7)

A spread of lines is a set of lines of P G(n, q), which partition the point set. A parallelism is a partition of the set of lines by spreads. A parallelism in P G(n, q) is point-cyclic if it has an automorphism group which acts as a Singer automorphism. In P G(3, 7) no point-cyclic parallelisms have been known. We claim that such do not exist. 1. Introduction. Parallelisms of projective spaces can be used to construct difference sets in connection with code synchronization [17]. Also parallelisms are used in constructions of constant dimension error-correcting codes that contain lifted maximum rank distance codes [7]. The relation of parallelisms to resolutions of Steiner systems leads to a cryptographic usage for anonymous (2, q + 1)-threshold schemes [15]. For the basic concepts and notations concerning spreads and parallelisms of projective spaces, refer, for instance, to [6], [10] or [16]. A t-spread in P G(n, q) is a set of distinct t-dimensional subspaces which partition the point set. A t-parallelism is a partition of the set of t-dimensional subspaces by t-spreads. Usually 1-spreads (1-parallelisms) are called line spreads (line parallelisms) or just spreads (parallelisms). Line spreads and parallelisms could exist if n is odd. Two parallelisms are isomorphic if there exists an automorphism of the projective space which maps each spread of the first parallelism to a spread of the second one. A subgroup of the automorphism group of the projective space which maps each spread of the parallelism to a spread of the same parallelism is called automorphism group of the parallelism. A parallelism is point-transitive if it has an automorphism group which is transitive on the points. If a parallelism has an automorphism which acts as a cycle of length equals to the number of points then it is point-cyclic.