Embeddings of Line-Grassmannians of Polar Spaces in Grassmann Varieties

Finite Fields and Their Applications

2018

Section: Hermitian Grassmannians and Their Embeddingsmentioning

confidence: 99%

Line Hermitian Grassmann codes and their parameters

Finite Fields and Their Applications

2018

“…For q even, an analogous argument, where we consider the absolute trace Tr 2 (q(A)) of q(A) instead of its quadratic character, leads to the same formula (7). = η 0 (q(A)) · q 2n−t .…”

Section: Even Tmentioning

confidence: 93%

Enumerative coding for line polar Grassmannians with applications to codes

Finite Fields and Their Applications

2017

A k-polar Grassmannian is a geometry having as pointset the set of all k-dimensional subspaces of a vector space V which are totally isotropic for a given non-degenerate bilinear form μ defined on V. Hence it can be regarded as a subgeometry of the ordinary k-Grassmannian. In this paper we deal with orthogonal line Grassmannians and with symplectic line Grassmannians, i.e. we assume k=2 and μ to be a non-degenerate symmetric or alternating form. We will provide a method to efficiently enumerate the pointsets of both orthogonal and symplectic line Grassmannians. This has several nice applications; among them, we shall discuss an efficient encoding/decoding/error correction strategy for line polar Grassmann codes of either type

“…Lemma 7.9. Let char(F) = 2 and let q : V → F be the generic non-degenerate quadratic form as given by (6). Then there exists a field extension F such that the extension q of q to V := V ⊗ F admits the following representation with respect to a suitable basis E of V :…”

Section: Weyl-like Embeddingsmentioning

confidence: 99%

Grassmann embeddings of polar Grassmannians

Journal of Combinatorial Theory, Series A

Pasini

2020

In this paper we compute the dimension of the Grassmann embeddings of the polar Grassmannians associated to a possibly degenerate Hermitian, alternating or quadratic form with possibly non-maximal Witt index. Moreover, in the characteristic 2 case, when the form is quadratic and non-degenerate with bilinearization of minimal Witt index, we define a generalization of the so-called Weyl embedding (see [4]) and prove that the Grassmann embedding is a quotient of this generalized 'Weyl-like' embedding. We also estimate the dimension of the latter.