111 research outputs found
The order of the automorphism group of a binary -analog of the Fano plane is at most two
It is shown that the automorphism group of a binary -analog of the Fano
plane is either trivial or of order .Comment: 10 page
New and Updated Semidefinite Programming Bounds for Subspace Codes
We show that and, more generally, by semidefinite programming for . Furthermore, we extend results by Bachoc et al. on SDP bounds for
, where is odd and is small, to for small and
small
A subspace code of size 333 in the setting of a binary q-analog of the Fano plane
We show that there is a binary subspace code of constant dimension 3 in
ambient dimension 7, having minimum distance 4 and cardinality 333, i.e., , which improves the previous best known lower bound of 329.
Moreover, if a code with these parameters has at least 333 elements, its
automorphism group is in one of conjugacy classes. This is achieved by a
more general technique for an exhaustive search in a finite group that does not
depend on the enumeration of all subgroups.Comment: 18 pages; typos correcte
Constructions of new matroids and designs over GF(q)
A perfect matroid design (PMD) is a matroid whose flats of the same rank all
have the same size. In this paper we introduce the q-analogue of a PMD and its
properties. In order to do that, we first establish a new cryptomorphic
definition for q-matroids. We show that q-Steiner systems are examples of
q-PMD's and we use this q-matroid structure to construct subspace designs from
q-Steiner systems. We apply this construction to S(2, 3, 13; q) q-Steiner
systems and hence establish the existence of subspace designs with previously
unknown parameters
q-analogs of group divisible designs
A well known class of objects in combinatorial design theory are {group
divisible designs}. Here, we introduce the -analogs of group divisible
designs. It turns out that there are interesting connections to scattered
subspaces, -Steiner systems, design packings and -divisible projective
sets.
We give necessary conditions for the existence of -analogs of group
divsible designs, construct an infinite series of examples, and provide further
existence results with the help of a computer search.
One example is a group divisible design over
which is a design packing consisting of blocks
that such every -dimensional subspace in is covered
at most twice.Comment: 18 pages, 3 tables, typos correcte
Coding Theory and Algebraic Combinatorics
This chapter introduces and elaborates on the fruitful interplay of coding
theory and algebraic combinatorics, with most of the focus on the interaction
of codes with combinatorial designs, finite geometries, simple groups, sphere
packings, kissing numbers, lattices, and association schemes. In particular,
special interest is devoted to the relationship between codes and combinatorial
designs. We describe and recapitulate important results in the development of
the state of the art. In addition, we give illustrative examples and
constructions, and highlight recent advances. Finally, we provide a collection
of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in
Information and Coding Theory", ed. by I. Woungang et al., World Scientific,
Singapore, 201
Generalized vector space partitions
A vector space partition in is a set of
subspaces such that every -dimensional subspace of is
contained in exactly one element of . Replacing "every point" by
"every -dimensional subspace", we generalize this notion to vector space
-partitions and study their properties. There is a close connection to
subspace codes and some problems are even interesting and unsolved for the set
case .Comment: 12 pages, typos correcte
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