Some partitions in Figueroa planes

Using Grundhöfer's construction of the Figueroa planes from Pappian planes  which  have an order $3$ planar collineation ${\widehat \alpha }$, we show that any  Figueroa plane (finite or infinite) has a partition of the complement of any proper (${\widehat \alpha }$)-invariant triangle mostly into subplanes together with a few  collinear  point sets (from the point set view) and a few concurrent line sets (from the  line set  view).  The partition shows that each Figueroa line (regarded as a set of  points) is  either the same as a Pappian line or consists mostly of a disjoint union of  subplanes of the Pappian plane (most lines are of this latter type) anddually. This last sentence is true with "Figueroa" and "Pappian" interchanged. There are many collinear subsets of Figueroa points which are a subset of the set of points of a Pappian conic and dually

Minimal symmetric differences of lines in projective planes

Let q be an odd prime power and let f(r) be the minimum size of the symmetric difference of r lines in the Desarguesian projective plane PG(2,q). We prove some results about the function f(r), in particular showing that there exists a constant C>0 such that f(r)=O(q) for Cq^{3/2}<r<q^2 - Cq^{3/2}.Comment: 16 pages + 2 pages of tables. This is a slightly revised version of the previous one (Thm 6 has been improved, and a few points explained

Finite linear spaces consisting of two symmetric configurations

We investigate finite linear spaces consisting of two symmetric configurations. A construction method using projective planes is presented, giving a possibly infinite number of examples. Other examples are constructed by difference families and automorphism groups, including a complete classification of the smallest case. A question whether any Steiner 2-design with twice as many lines as points belongs to this family of linear spaces is raised, and answered in the affirmative for all known examples of such designs

LDPC codes from the Hermitian curve

In this paper, we study the code C which has as parity check matrix H the incidence matrix of the design of the Hermitian curve and its (q + 1)-secants. This code is known to have good performance with an iterative decoding algorithm, as shown by Johnson and Weller in ( Proceedings at the ICEE Globe com conference, Sanfrancisco, CA, 2003). We shall prove that C has a double cyclic structure and that by shortening in a suitable way H it is possible to obtain new codes which have higher code-rate. We shall also present a simple way to constructing the matrix H via a geometric approach