5 research outputs found

    On transparent embeddings of point-line geometries

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    We introduce the class of transparent embeddings for a point-line geometry Γ=(P,L)\Gamma = ({\mathcal P},{\mathcal L}) as the class of full projective embeddings ε\varepsilon of Γ\Gamma such that the preimage of any projective line fully contained in ε(P)\varepsilon({\mathcal P}) is a line of Γ\Gamma. We will then investigate the transparency of Pl\"ucker embeddings of projective and polar grassmannians and spin embeddings of half-spin geometries and dual polar spaces of orthogonal type. As an application of our results on transparency, we will derive several Chow-like theorems for polar grassmannians and half-spin geometries.Comment: 28 Pages/revised version after revie

    Grassmann embeddings of polar Grassmannians

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    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 22 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 [I. Cardinali and A. Pasini, Grassmann and Weyl embeddings of orthogonal Grassmannians. J. Algebr. Combin. 38 (2013), 863-888]) and prove that the Grassmann embedding is a quotient of this generalized "Weyl-like" embedding. We also estimate the dimension of the latter.Comment: 25 pages/revised version after revie

    Grassmannians of codes

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    Consider the point line-geometry Pt(n,k){\mathcal P}_t(n,k) having as points all the [n,k][n,k]-linear codes having minimum dual distance at least t+1t+1 and where two points XX and YY are collinear whenever X∩YX\cap Y is a [n,k−1][n,k-1]-linear code having minimum dual distance at least t+1t+1. We are interested in the collinearity graph Λt(n,k)\Lambda_t(n,k) of Pt(n,k).{\mathcal P}_t(n,k). The graph Λt(n,k)\Lambda_t(n,k) is a subgraph of the Grassmann graph and also a subgraph of the graph Δt(n,k)\Delta_t(n,k) of the linear codes having minimum dual distance at least t+1t+1 introduced in~[M. Kwiatkowski, M. Pankov, On the distance between linear codes, Finite Fields Appl. 39 (2016), 251--263, doi:https://doi.org/10.1016/j.ffa.2016.02.004, arXiv:1506.00215]. We shall study the structure of Λt(n,k)\Lambda_t(n,k) in relation to that of Δt(n,k)\Delta_t(n,k) and we will characterize the set of its isolated vertices. We will then focus on Λ1(n,k)\Lambda_1(n,k) and Λ2(n,k)\Lambda_2(n,k) providing necessary and sufficient conditions for them to be connected

    Line Hermitian Grassmann codes and their parameters

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    In 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
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