3,508 research outputs found
Subspace Polynomials and Cyclic Subspace Codes
Subspace codes have received an increasing interest recently due to their
application in error-correction for random network coding. In particular,
cyclic subspace codes are possible candidates for large codes with efficient
encoding and decoding algorithms. In this paper we consider such cyclic codes
and provide constructions of optimal codes for which their codewords do not
have full orbits. We further introduce a new way to represent subspace codes by
a class of polynomials called subspace polynomials. We present some
constructions of such codes which are cyclic and analyze their parameters
Stabilizer codes from modified symplectic form
Stabilizer codes form an important class of quantum error correcting codes
which have an elegant theory, efficient error detection, and many known
examples. Constructing stabilizer codes of length is equivalent to
constructing subspaces of which are
"isotropic" under the symplectic bilinear form defined by . As a
result, many, but not all, ideas from the theory of classical error correction
can be translated to quantum error correction. One of the main theoretical
contribution of this article is to study stabilizer codes starting with a
different symplectic form.
In this paper, we concentrate on cyclic codes. Modifying the symplectic form
allows us to generalize the previous known construction for linear cyclic
stabilizer codes, and in the process, circumvent some of the Galois theoretic
no-go results proved there. More importantly, this tweak in the symplectic form
allows us to make use of well known error correcting algorithms for cyclic
codes to give efficient quantum error correcting algorithms. Cyclicity of error
correcting codes is a "basis dependent" property. Our codes are no more
"cyclic" when they are derived using the standard symplectic forms (if we
ignore the error correcting properties like distance, all such symplectic forms
can be converted to each other via a basis transformation). Hence this change
of perspective is crucial from the point of view of designing efficient
decoding algorithm for these family of codes. In this context, recall that for
general codes, efficient decoding algorithms do not exist if some widely
believed complexity theoretic assumptions are true
Cyclic Orbit Codes
In network coding a constant dimension code consists of a set of
k-dimensional subspaces of F_q^n. Orbit codes are constant dimension codes
which are defined as orbits of a subgroup of the general linear group, acting
on the set of all subspaces of F_q^n. If the acting group is cyclic, the
corresponding orbit codes are called cyclic orbit codes. In this paper we give
a classification of cyclic orbit codes and propose a decoding procedure for a
particular subclass of cyclic orbit codes.Comment: submitted to IEEE Transactions on Information Theor
A Complete Characterization of Irreducible Cyclic Orbit Codes and their Pl\"ucker Embedding
Constant dimension codes are subsets of the finite Grassmann variety. The
study of these codes is a central topic in random linear network coding theory.
Orbit codes represent a subclass of constant dimension codes. They are defined
as orbits of a subgroup of the general linear group on the Grassmannian. This
paper gives a complete characterization of orbit codes that are generated by an
irreducible cyclic group, i.e. a group having one generator that has no
non-trivial invariant subspace. We show how some of the basic properties of
these codes, the cardinality and the minimum distance, can be derived using the
isomorphism of the vector space and the extension field. Furthermore, we
investigate the Pl\"ucker embedding of these codes and show how the orbit
structure is preserved in the embedding.Comment: submitted to Designs, Codes and Cryptograph
A Complete Characterization of Irreducible Cyclic Orbit Codes
We give a complete list of orbit codes that are generated by an irreducible
cyclic group, i.e. an irreducible group having one generator. We derive some of
the basic properties of these codes such as the cardinality and the minimum
distance.Comment: in Proceedings of The Seventh International Workshop on Coding and
Cryptography 2011 April 11-15 2011, Paris, Franc
- …