Line polar Grassmann codes of orthogonal type

Abstract

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

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.…”

“…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 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].…”

Implementing line-Hermitian Grassmann codes

“…(E6) a t = b t = 0 and A t = 0 = B t . By System (7), the set of lines of H m with (even) prefix D t = A t B t is the same as the disjoint union of the sets of lines…”