Arcs in finite projective spaces

Arcs in Finite Projective Spaces

These notes are an outline of a course on arcs given at the Finite Geometry Summer School, University of Sussex, June 26-30, 2017. Basic objects and definitions Let K denote an arbitrary field. Let Fq denote the finite field with q elements, where q is the power of a prime p. Let Vk(K) denote the k-dimensional vector space over K. Let PGk−1(K) denote the (k − 1)-dimensional projective space ove...

Maximum distance separable codes and arcs in projective spaces

Given any linear code C over a finite field GF(q) we show how C can be described in a transparent and geometrical way by using the associated Bruen–Silverman code. Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing...

The dual construction for arcs in projective Hjelmslev spaces

On multiple caps in finite projective spaces

In this paper, we consider new results on (k, n)-caps with n > 2. We provide a lower bound on the size of such caps. Furthermore, we generalize two product constructions for (k, 2)-caps to caps with larger n. We give explicit constructions for good caps with small n. In particular, we determine the largest size of a (k, 3)-cap in PG(3, 5), which turns out to be 44. The results on caps in PG(3, ...

Partial k-Parallelisms in Finite Projective Spaces

In this paper, we consider the following question. What is the maximum number of pairwise disjoint k-spreads that exist in PG(n, q)? We prove that if k + 1 divides n + 1 and n > k then there exist at least two disjoint k-spreads in PG(n, q) and there exist at least 2k+1 − 1 pairwise disjoint k-spreads in PG(n, 2). We also extend the known results on parallelism in a projective geometry from whi...