8 research outputs found
On the Geometry of Balls in the Grassmannian and List Decoding of Lifted Gabidulin Codes
The finite Grassmannian is defined as the set of all
-dimensional subspaces of the ambient space . Subsets of
the finite Grassmannian are called constant dimension codes and have recently
found an application in random network coding. In this setting codewords from
are sent through a network channel and, since errors may
occur during transmission, the received words can possible lie in
, where . In this paper, we study the balls in
with center that is not necessarily in
. We describe the balls with respect to two different
metrics, namely the subspace and the injection metric. Moreover, we use two
different techniques for describing these balls, one is the Pl\"ucker embedding
of , and the second one is a rational parametrization of
the matrix representation of the codewords.
With these results, we consider the problem of list decoding a certain family
of constant dimension codes, called lifted Gabidulin codes. We describe a way
of representing these codes by linear equations in either the matrix
representation or a subset of the Pl\"ucker coordinates. The union of these
equations and the equations which arise from the description of the ball of a
given radius in the Grassmannian describe the list of codewords with distance
less than or equal to the given radius from the received word.Comment: To be published in Designs, Codes and Cryptography (Springer
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
Equidistant Codes in the Grassmannian
Equidistant codes over vector spaces are considered. For -dimensional
subspaces over a large vector space the largest code is always a sunflower. We
present several simple constructions for such codes which might produce the
largest non-sunflower codes. A novel construction, based on the Pl\"{u}cker
embedding, for 1-intersecting codes of -dimensional subspaces over \F_q^n,
, where the code size is is
presented. Finally, we present a related construction which generates
equidistant constant rank codes with matrices of size
over \F_q, rank , and rank distance .Comment: 16 page