602 research outputs found
Minimum distance of Orthogonal Line-Grassmann Codes in even characteristic
No full textIn this paper we determine the minimum distance of orthogonal line-Grassmann codes for q even. The case q odd was solved in "I. Cardinali, L. Giuzzi, K. Kaipa, A. Pasini, Line Polar Grassmann Codes of Orthogonal Type, J. Pure Applied Algebra (doi:10.1016/j.jpaa.2015.10.007 )" We also show that for q even all minimum weight codewords are equivalent and that symplectic line-Grassmann codes are proper subcodes of codimension 2n of the orthogonal ones
On orthogonal polar spaces
No full textLet be a non-degenerate polar space. In [I. Cardinali, L. Giuzzi, A.
Pasini, "The generating rank of a polar grassmannian", Adv. Geom. 21:4 (2021),
515-539 doi:10.1515/advgeom-2021-0022 (arXiv:1906.10560)] we introduced an
intrinsic parameter of , called the anisotropic gap, defined as the
least upper bound of the lengths of the well-ordered chains of subspaces of
containing a frame; when is orthogonal, we also defined two
other parameters of , called the elliptic and parabolic gap, related to
the universal embedding of .
In this paper, assuming is an orthogonal polar space, we prove that
the elliptic and parabolic gaps can be described as intrinsic invariants of
without making recourse to the embedding.Comment: 20 pages/revised versio
An outline of polar spaces: basics and advances
No full textThis paper is an extended version of a series of lectures on polar spaces given during the workshop and conference \lq Groups and Geometries\rq\, held at the Indian Statistical Institute in Bangalore in December 2012.
We firstly give a concise exposition of the theory of polar spaces, ending up with the classification of polar spaces of rank at least . Then we present a few related research topics, as polar spaces of infinite rank, non-linear embeddings of polar spaces, projective embeddings of dual polar spaces and polar grassmannians
Line polar Grassmann codes of orthogonal type
No full textPolar Grassmann codes of orthogonal type have been introduced in I. Cardinali and L. Giuzzi, \emph{Codes and caps from orthogonal Grassmannians}, {Finite Fields Appl.} {\bf 24} (2013), 148-169. They are subcodes of the Grassmann code arising from the projective system defined by the Pl\"ucker embedding of a polar Grassmannian of orthogonal type. In the present paper we fully determine the minimum distance of line polar Grassmann Codes of orthogonal type for q odd
Non-projective embeddings in the grassmann variety
No full texte investigate properties of the grassmann embedding of dual classical thick generalized quadrangles focusing on the grassmann embedding of the dual of an orthogonal quadrangle and the dual of a hermitian quadrangle We prove that, if the characteristic of the field is different from 2 then the dimension of the grassmann embedding of is and its image is isomorphic to the quadratic veronese variety of a 3-dimensional projective space. If is a perfect field of characteristic 2 then the dimension of the grassmann embedding of is proved to be and its image is a -dimensional algebraic subvariety of the grassmannian of lines of a -dimensional projective space.
Moving to consider the dual quadrangle , we prove that the dimension of its grassmann embedding is and the image of under the grassmann embedding is a -dimensional algebraic subvariety of the grassmannian of lines of a -dimensional projective space
Codes and caps from orthogonal Grassmannians
No full textIn this paper we investigate linear error correcting codes and
projective caps related to the Grassmann embedding of an
orthogonal Grassmannian. In particular, we determine some of
the parameters of the codes arising from the projective system
determined by such an embedding. We also study special sets of points of
which are met by any line of k in subsets of cardinality at most 2 and
we show that their image under the Grassmann embedding is
a projective cap
Line Polar Grassmann Codes of Orthogonal type
No full textPolar Grassmann codes of orthogonal type have been introduced in I. Cardinali and L. Giuzzi, \emph{Codes and caps from orthogonal Grassmannians}, {Finite Fields Appl.} {\bf 24} (2013), 148-169. They are subcodes of the Grassmann code arising from the projective system defined by the Pl\"ucker embedding of a polar Grassmannian of orthogonal type. In the present paper we fully determine the minimum distance of line polar Grassmann Codes of orthogonal type for q odd
Enumerative coding for line polar Grassmannians with applications to codes
No full textA 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
Line Hermitian Grassmann codes and their parameters
No full textIn this paper we introduce and study line Hermitian Grassmann codes as those subcodes of the Grassmann codes associated to the 2-Grassmannian of a Hermitian polar space defined over a finite field. In particular, we determine the parameters and characterize the words of minimum weight
The q-clan geometries with q=2e
No full textLa monografia rappresenta una trattazione coerente e completa della teoria dei quadrangoli generalizzati associati a flock in caratteristica 2 fornendo una descrizione delle principale strutture geometriche ad essi associate e riportandone i risultati piu' significativi
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