2,297 research outputs found
Johnson type bounds for mixed dimension subspace codes
Subspace codes, i.e., sets of subspaces of , are applied in
random linear network coding. Here we give improved upper bounds for their
cardinalities based on the Johnson bound for constant dimension codes.Comment: 16 pages, typos correcte
Tables of subspace codes
One of the main problems of subspace coding asks for the maximum possible
cardinality of a subspace code with minimum distance at least over
, where the dimensions of the codewords, which are vector
spaces, are contained in . In the special case of
one speaks of constant dimension codes. Since this (still) emerging
field is very prosperous on the one hand side and there are a lot of
connections to classical objects from Galois geometry it is a bit difficult to
keep or to obtain an overview about the current state of knowledge. To this end
we have implemented an on-line database of the (at least to us) known results
at \url{subspacecodes.uni-bayreuth.de}. The aim of this recurrently updated
technical report is to provide a user guide how this technical tool can be used
in research projects and to describe the so far implemented theoretic and
algorithmic knowledge.Comment: 44 pages, 6 tables, 7 screenshot
Subspace code constructions
We improve on the lower bound of the maximum number of planes of mutually intersecting in at most one point leading to the following
lower bound: for
constant dimension subspace codes. We also construct two new non-equivalent
constant dimension subspace orbit-codes
Optimal Binary Subspace Codes of Length 6, Constant Dimension 3 and Minimum Distance 4
It is shown that the maximum size of a binary subspace code of packet length
, minimum subspace distance , and constant dimension is ;
in Finite Geometry terms, the maximum number of planes in
mutually intersecting in at most a point is .
Optimal binary subspace codes are classified into
isomorphism types, and a computer-free construction of one isomorphism type is
provided. The construction uses both geometry and finite fields theory and
generalizes to any , yielding a new family of -ary
subspace codes
Subspace Packings : Constructions and Bounds
The Grassmannian is the set of all -dimensional
subspaces of the vector space . K\"{o}tter and Kschischang
showed that codes in Grassmannian space can be used for error-correction in
random network coding. On the other hand, these codes are -analogs of codes
in the Johnson scheme, i.e., constant dimension codes. These codes of the
Grassmannian also form a family of -analogs of block
designs and they are called subspace designs. In this paper, we examine one of
the last families of -analogs of block designs which was not considered
before. This family, called subspace packings, is the -analog of packings,
and was considered recently for network coding solution for a family of
multicast networks called the generalized combination networks. A subspace
packing - is a set of -subspaces from
such that each -subspace of is
contained in at most elements of . The goal of this work
is to consider the largest size of such subspace packings. We derive a sequence
of lower and upper bounds on the maximum size of such packings, analyse these
bounds, and identify the important problems for further research in this area.Comment: 30 pages, 27 tables, continuation of arXiv:1811.04611, typos
correcte
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