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A Bruen chain forq=19

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Abstract

The one-to-one correspondence between the class of two-dimensional translation planes of orderq 2 and the collection of spreads ofPG(3,q) has long provided a natural context for describing new planes. The method often used for constructing “interesting” spreads is to start with a regular spread, corresponding to a desarguesian plane, and then replace some “nice” subset of lines by another partial spread covering the same set of points. Indeed the first approach was replacing the lines of a regulus by the lines of its opposite regulus, or doing this process for a set of disjoint reguli. Nontrivial generalizations of this idea include thechains of Bruen and thenests of Baker and Ebert. In this paper we construct a replaceable subset of a regular spread ofPG (3, 19) which is the union of 11 reguli double covering the lines in their union, hence is a chain in the terminology of Bruen or a 11-nest in the Baker-Ebert terminology.

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Baker, R.D., Ebert, G.L. A Bruen chain forq=19. Des Codes Crypt 4, 307–312 (1994). https://doi.org/10.1007/BF01388646

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