Codes and caps from orthogonal Grassmannians

Abstract

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.

Minimum distance of orthogonal line-Grassmann codes in even characteristic

“…Using the aforementioned results of [1,3] this leads to the following general result. Note that for q odd and n = 2, by [3,Corollary 3.8], the minimum weight codewords lie on two orbits under the action of the linear automorphism group of the code.…”

“…A more difficult task is to determine the minimum distance of an orthogonal Grassmann code. In [1] we obtained the exact value of d min for n = k = 2 and n = k = 3; more recently, in [3], it has been shown that for q odd and k = 2 the minimum distance of P n,2 is q 4n−5 − q 3n−4 . We now present in detail a geometric setting in which it is possible to study the weights of a projective code arising from the image under the Plücker embedding ε k of an arbitrary set of k-subspaces.…”

“…The codes P n,k := C(Ω) are called orthogonal k-Grassmann codes and they were introduced in [1]. Theorem 2.1 immediately provides the length N as the number of points of ∆ k and the dimension K = dim W k of P n,k .…”

“…The codes associated with polar k-Grassmannians of either orthogonal, symplectic or Hermitian type have been introduced respectively in [1], [4] and [5]. In the case of line-Grassmannians, that is for k = 2, the following results are known: in the symplectic case it has been shown in [4] that the minimum distance is q 4n−5 − q 2n−3 for any q; in the orthogonal case it has been shown in [1,Main Result 2] that the minimum distance is d min = q 3 − q 2 for n = 2 for any q and in [3] that the minimum distance is d min = q 4n−5 − q 3n−4 for q odd. Similar results hold in the Hermitian case, see [5].…”

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

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AbstractIn 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 ε gr k of an orthogonal Grassmannian ∆ k . More precisely, we consider the codes arising from the projective system determined by ε gr k (∆ k ) and determine some of their parameters.

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