OBJECTS IN AFFINE AND PROJECTIVE SPACES
Galois geometries are well known to be rich of nice geometric and combinatorial properties that have also found wide and relevant applications in coding theory.
CAPS. An important issue in this context is to ask for explicit constructions of small complete caps. A cap in a Galois space is a set of points no three of which are collinear; a cap is said to be complete if it is maximal with respect to set-theoretical inclusion. From complete caps there arise linear codes which turn out to have good covering properties, provided that the size of the cap is small with respect to the dimension N and the order q of the ambient space. The trivial lower bound for the size of such a complete cap is √2q^((N-1)/2). If q is even and N is odd, this bound is substantially sharp; otherwise, all known infinite families of complete caps have size far from the trivial lower bound.
In [2013/4] small complete caps in a three-dimensional Galois space of odd order q are investigated. In [2012/2] caps in PG(4,4) with extra properties (quatum caps) are classified. Quantum caps are further studied in [2014/8]. In [2011/1] the minimum order of complete caps in PG(4,4) is determined via a computer based search. In [2014/2] [2015/3] [2017/10] small complete caps in affine spaces are obtained in connection with so called bicovering arcs from cubic plane curves. In [2017/3] a probabilistic method for constructing small complete arcs in projective planes is exented to small complete caps in projective spaces. In [2017/5] [2017/7] [2019/6] computer results on small complete caps in projective spaces of dimension 3 and 4 are presented. Small complete caps in affine spaces over fields of characteristic 2 are obtained via a connection via with maximum scattered linear sets in [2018/5]. Conjectural upper bounds on the smallest size of a complete cap in projectives space are investigated in [2017/6].