Doubly \({\beta}\)-derived translation planes
Let \(q>3\) be an odd prime power. A chain of circles in the Miquelian inversive plane \(M(q)\) is a collection of \({1\over 2}(q+3)\) circles such that every point covered by some circle in this set lies on exactly two circles from the collection. Using the correspondence between points and circles of \(M(q)\) and lines and reguli of a regular spread \(S\) in \(PG(3,q)\), it is shown in [\textit{O. Heden}, Discrete Math. 146, 69-96 (1995; Zbl 0840.51002)] that the resulting set of \({1 \over 4} (q+3)(q+1)\) lines in \(S\) is always replaceable in the sense of Ostrom's net replacement. For \(q = 7\) there are precisely two such chains, up to projective equivalence, which result in two non-Desarguesian translation planes of order 49 by net replacement. In the paper under review an exhaustive computer search is used to classify those translation planes of order 49 which can be obtained by (multiple) derivation (net replacement of a regulus by its opposite regulus) from one of the above two planes. It is shown that, up to isomorphism, there are precisely five such planes, three of which previously appeared in print. It should be noted that in [\textit{R. Mathon} and \textit{G. F. Royle}, Des. Codes Cryptography 5, 57-72 (1995; Zbl 0819.51003)] all translation planes of order 49 are enumerated. However, the data given in that enumeration is not sufficient for solving the classification problem discussed in this paper.
scientific article; zbMATH DE number 1892219