Translation planes and derivation sets (Q917909)

The authors continue their work on coding theoretic techniques applied to finite geometry. Let \(\pi\) be an affine plane of order n, and let p be a prime dividing n. Let \(N=n^ 2\), \(F_ p\) the finite field of order p, and \(F^ N_ p\) the vector space of all functions from the point set of \(\pi\) to \(F_ p\). Let \(C_ p(\pi)\) be the subspace of \(F^ N_ p\) generated by the characteristic functions of the lines of \(\pi\). If \(\pi\) is a translation plane, then certainly \(n=q=p^ s\) for some prime p and some positive integer s. Let \(K=F_ q\) and view \(V=K\oplus K\) as a 2s dimensional vector space over \(F_ q\). Let \(F^ V_ p\) be the vector space of all functions from V to \(F_ p\). Let F be any subfield of K, and let B(K\(| F)\) be the subspace of \(F^ V_ p\) generated by all characteristic vectors of F-flats (s-dimensional flats of V that are cosets of F-subspaces of V). Say that \(\pi\) is ``contained'' in B(K\(| F)\) if and only if \(C_ p(\pi)\) is isomorphic to a subcode of B(K\(| F)\). This equivalent to saying \(\pi\) has kernel F in the usual geometric sense. Finally, let E(K\(| F)\) be the subspace of \(F^ V_ p\) generated by all differences of characteristic vectors \(v^ X-v^ Y\), where X and Y are parallel F-flats of V. The authors use this machinery to give a new upper bound for the p-rank of \(\pi\) ; namely, they show \(\dim (C_ p(\pi))\leq q+\dim (E(K| F))\) whenever \(\pi\) is a translation plane contained in B(K\(| F)\) as above. However, the main purpose of the paper is to give a general notion of derivation set for any finite projective plane by using coding theoretic ideas similar to those above. The ``classical'' derivation uses a Baer segment D on the line L at infinity and a collection of Baer subplanes having L as a line such that D is the intersection of L with each of these Baer subplanes. The authors show that for translation planes the ``classical'' Baer subplane derivation takes place in B(K\(| F)\) as defined above. Specializing to \(p=2\) and letting \(q=2^ s\), it is then shown how a set of ovals in a translation plane coordinatized by a nearfield of order q can be used to define a derivation set (in the coding theoretic sense) of cardinality q-1. Other kinds of derivation sets are also considered.