Laguerre planes of even order and translation ovals
The paper illustrates how translation planes of even order \(q=2^ s\) can be represented in the projective space \(\text{PG}(2s-1,2)\) by a collection of \(q^ 2+1\) subspaces of dimension \(s-1\) satisfying a rather simple condition. A translation Laguerre plane, or a Laguerre plane of Kleinewillinghöfer type at least I, is a Laguerre plane with a distinct parallel class \(\pi_ \infty\) such that the internal structure at any point of \(\pi_ \infty\) is a translation plane and such that any translation in such a translation plane is induced by an automorphism of the Laguerre plane. If the Laguerre plane is of even order \(q=2^ s\) then every circle can be viewed as a translation oval and thus corresponds to an \(s\)- dimensional subspace of the \(2s\)-dimensional vector space over \(\text{GF}(2)\). Also parallel classes correspond to \(s\)-dimensional subspaces of the \(2s\)-dimensional vector space over \(\text{GF}(2)\). In the \(2s- 1\) dimensional space at infinity \(\mathcal P\) these \(s\)- dimensional subspaces induce a collection \({\mathcal S}=\{S_ 1,\dots, S_{q^ 2}, S_ \infty\}\) of \(q^ 2+1\) subspaces of dimension \(s-1\) such that: (1) \(s_ i\cap S_ \infty=\emptyset\) for \(1\leq i\leq q^ 2\) (2) Every line of \({\mathcal P}\) skew to \(S_ \infty\) is contained in precisely one of the subspaces \(S_ i\), \(1\leq i\leq q^ 2\). Conversely, any collection \({\mathcal S}=\{S_ 1, \dots, S_{q^ 2}, S_ \infty\}\) of \(q^ 2+1\) subspaces of dimension \(s-1\) of \(\text{PG}(2s-1,2)\) satisfying these two conditions can be constructed from a translation Laguerre plane of order \(2^ s\). In fact, any such collection gives rise to two Laguerre planes. To construct the Laguerre plane from the collection \(\mathcal S\) of subspaces one needs a collection of \(q\) spreads \(\Sigma_ 1, \dots, \Sigma_ q\) of \(\text{PG}(2s- 1,2)\) contained in \(\mathcal S\) such that any two different spreads have precisely the subspace \(S_ \infty\) in common. Using results on dual nets one sees that there are always two such collections \(\{\Sigma_ 1, \dots, \Sigma_ q\}\) and \(\{\Sigma_ 1'. \dots, \Sigma_ q'\}\) of spreads where, in addition, \(|\Sigma_ i\cap \Sigma_ j|\leq 2\) for \(1\leq i, j \leq q\). Translating the main result into the language of Linear Algebra the authors obtain that a translation plane of order \(q=2^ s\) yields a collection of \(q^ 2\) quadratic \(s\times s\)-matrices over \(\text{GF}(2)\) such that the rank of the sum of any pair of these matrices is at least \(s-1\). The paper closes with an illustration of the results by looking at finite ovoidal translation Laguerre planes of even order.
scientific article; zbMATH DE number 663752
scientific article