6 research outputs found
Lifted codes and the multilevel construction for constant dimension codes
Constant dimension codes are e.g. used for error correction and detection in
random linear network coding, so that constructions for these codes have
achieved wide attention. Here, we improve over 150 lower bounds by describing
better constructions for subspace distance 4.Comment: 40 pages, 8 table
The interplay of different metrics for the construction of constant dimension codes
A basic problem for constant dimension codes is to determine the maximum
possible size of a set of -dimensional subspaces in
, called codewords, such that the subspace distance satisfies
for all pairs of different codewords ,
. Constant dimension codes have applications in e.g.\ random linear network
coding, cryptography, and distributed storage. Bounds for are the
topic of many recent research papers. Providing a general framework we survey
many of the latest constructions and show up the potential for further
improvements. As examples we give improved constructions for the cases
, , , and . We also derive
general upper bounds for subcodes arising in those constructions.Comment: 19 pages; typos correcte
Bounds for the multilevel construction
One of the main problems in random network coding is to compute good lower
and upper bounds on the achievable cardinality of the so-called subspace codes
in the projective space for a given minimum distance. The
determination of the exact maximum cardinality is a very tough discrete
optimization problem involving a huge number of symmetries. Besides some
explicit constructions for \textit{good} subspace codes several of the most
success full constructions involve the solution of discrete optimization
subproblems itself, which mostly have not been not been solved systematically.
Here we consider the multilevel a.k.a.\ Echelon--Ferrers construction and given
lower and upper bounds for the achievable cardinalities. From a more general
point of view, we solve maximum clique problems in weighted graphs, where the
weights can be polynomials in the field size .Comment: 95 page