14 research outputs found
Projective divisible binary codes
For which positive integers does there exist a linear code
over with all codeword weights divisible by and such
that the columns of a generating matrix of are projectively distinct? The
motivation for studying this problem comes from the theory of partial spreads,
or subspace codes with the highest possible minimum distance, since the set of
holes of a partial spread of -flats in
corresponds to a -divisible code with . In this paper we provide
an introduction to this problem and report on new results for .Comment: 10 pages, 3 table
On the non-existence of a projective (75, 4,12, 5) set in PG(3, 7)
We show by a combination of theoretical argument and computer search that if a projective (75, 4, 12, 5) set in PG(3, 7) exists then its automorphism group must be trivial. This corresponds to the smallest open case of a coding problem posed by H. Ward in 1998, concerning the possible existence of an infinite family of projective two-weight codes meeting the Griesmer bound
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