Implementing Line-Hermitian Grassmann codes

In [I. Cardinali and L. Giuzzi. Line Hermitian Grassmann codes and their parameters. Finite Fields Appl., 51: 407-432, 2018] we introduced line Hermitian Grassmann codes and determined their parameters. The aim of this paper is to present (in the spirit of [I. Cardinali and L. Giuzzi. Enumerative coding for line polar Grassmannians with applications to codes. Finite Fields Appl., 46:107-138, 2017]) an algorithm for the point enumerator of a line Hermitian Grassmannian which can be usefully applied to get efficient encoders, decoders and error correction algorithms for the aforementioned codes.Comment: 26 page

Minimum distance of Symplectic Grassmann codes

We introduce the Symplectic Grassmann codes as projective codes defined by symplectic Grassmannians, in analogy with the orthogonal Grassmann codes introduced in [4]. Note that the Lagrangian-Grassmannian codes are a special class of Symplectic Grassmann codes. We describe the weight enumerator of the Lagrangian--Grassmannian codes of rank $2$ and $3$ and we determine the minimum distance of the line Symplectic Grassmann codes.Comment: Revised contents and biblograph

Line Polar Grassmann Codes of Orthogonal Type

Polar 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

Some results on caps and codes related to orthogonal Grassmannians — a preview

In this note we offer a short summary of some recent results, to be contained in a forthcoming paper [4], on projective caps and linear error correcting codes arising from the Grassmann embedding εgr k of an orthogonal Grassmannian ∆k . More precisely, we consider the codes arising from the projective system determined by εgr k (∆k ) and determine some of their parameters. We also investigate special sets of points of ∆k which are met by any line of ∆k in at most 2 points proving that their image under the Grassmann embedding is a projective cap

Enumerative Coding for Line Polar Grassmannians with applications to codes

A $k$-polar Grassmannian is the 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 $\mu$ 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 $\mu$ 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 both types.Comment: 27 pages; revised version after revie