1,405 research outputs found
Decomposable Subspaces, Linear Sections of Grassmann Varieties, and Higher Weights of Grassmann Codes
Given a homogeneous component of an exterior algebra, we characterize those
subspaces in which every nonzero element is decomposable. In geometric terms,
this corresponds to characterizing the projective linear subvarieties of the
Grassmann variety with its Plucker embedding. When the base field is finite, we
consider the more general question of determining the maximum number of points
on sections of Grassmannians by linear subvarieties of a fixed (co)dimension.
This corresponds to a known open problem of determining the complete weight
hierarchy of linear error correcting codes associated to Grassmann varieties.
We recover most of the known results as well as prove some new results. In the
process we obtain, and utilize, a simple generalization of the Griesmer-Wei
bound for arbitrary linear codes.Comment: 16 page
Hyperplanes of Hermitian dual polar spaces of rank 3 containing a quad
Let F and F' be two fields such that F' is a quadratic Galois extension of F. If vertical bar F vertical bar >= 3, then we provide sufficient conditions for a hyperplane of the Hermitian dual polar space DH(5, F') to arise from the Grassmann embedding. We use this to give an alternative proof for the fact that all hyperplanes of DH(5, q(2)), q not equal 2, arise from the Grassmann embedding, and to show that every hyperplane of DH(5, F') that contains a quad Q is either classical or the extension of a non-classical ovoid of Q. We will also give a classification of the hyperplanes of DH(5, F') that contain a quad and arise from the Grassmann embedding
Points and hyperplanes of the universal embedding space of the dual polar space DW(5,q), q odd
It was proved earlier that there are 6 isomorphism classes of hyperplanes in the dual polar space (5,q) even, which arise from its Grassmann-embedding. In the present paper, we extend these results to the case that , which affords the universal embedding of (5,q)$
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
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