13 research outputs found
The Classification of the Largest Caps in AG(5, 3)
AbstractWe prove that 45 is the size of the largest caps in AG(5,3), and such a 45-cap is always obtained from the 56-cap in PG(5,3) by deleting an 11-hyper-plane
Good Random Matrices over Finite Fields
The random matrix uniformly distributed over the set of all m-by-n matrices
over a finite field plays an important role in many branches of information
theory. In this paper a generalization of this random matrix, called k-good
random matrices, is studied. It is shown that a k-good random m-by-n matrix
with a distribution of minimum support size is uniformly distributed over a
maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and
vice versa. Further examples of k-good random matrices are derived from
homogeneous weights on matrix modules. Several applications of k-good random
matrices are given, establishing links with some well-known combinatorial
problems. Finally, the related combinatorial concept of a k-dense set of m-by-n
matrices is studied, identifying such sets as blocking sets with respect to
(m-k)-dimensional flats in a certain m-by-n matrix geometry and determining
their minimum size in special cases.Comment: 25 pages, publishe
Partial spreads and vector space partitions
Constant-dimension codes with the maximum possible minimum distance have been
studied under the name of partial spreads in Finite Geometry for several
decades. Not surprisingly, for this subclass typically the sharpest bounds on
the maximal code size are known. The seminal works of Beutelspacher and Drake
\& Freeman on partial spreads date back to 1975, and 1979, respectively. From
then until recently, there was almost no progress besides some computer-based
constructions and classifications. It turns out that vector space partitions
provide the appropriate theoretical framework and can be used to improve the
long-standing bounds in quite a few cases. Here, we provide a historic account
on partial spreads and an interpretation of the classical results from a modern
perspective. To this end, we introduce all required methods from the theory of
vector space partitions and Finite Geometry in a tutorial style. We guide the
reader to the current frontiers of research in that field, including a detailed
description of the recent improvements.Comment: 30 pages, 1 tabl