In this paper we investigate linear error correcting codes and projective caps related to the Grassmann embedding ε gr k of an orthogonal Grassmannian k . In particular, we determine some of the parameters of the codes arising from the projective system determined by ε gr k ( k ). We also study special sets of points of k which are met by any line of k in at most 2 points and we show that their image under the Grassmann embedding ε gr k is a projective cap.
Polar Grassmann codes of orthogonal type have been introduced in [1]. They are punctured versions of the Grassmann code arising from the projective system defined by the Plücker 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
For any admissible value of the parameters n and k there exist [n, k]-Maximum Rank Distance F q -linear codes. Indeed, it can be shown that if field extensions large enough are considered, almost all rank metric codes are MRD. On the other hand, very few families up to equivalence of such codes are currently known. In the present paper we study some invariants of MRD codes and evaluate their value for the known families, providing a new characterization of generalized twisted Gabidulin codes.
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
In this paper we introduce symplectic Grassmann codes, in analogy to ordinary Grassmann codes and orthogonal Grassmann codes, as projective codes defined by symplectic Grassmannians. Lagrangian-Grassmannian codes are a special class of symplectic Grassmann codes. We describe all the parameters of line symplectic Grassmann codes and we provide the full weight enumerator for the Lagrangian-Grassmannian codes of rank 2 and 3
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