Minimum distance of orthogonal line-Grassmann codes in even characteristic

Abstract

In this paper we determine the minimum distance of orthogonal line-Grassmann codes for q even. The case q odd was solved in [3]. For n = 3 we also determine the second smallest distance. Furthermore, we show that for q even all minimum weight codewords are equivalent and that symplectic line-Grassmann codes are proper subcodes of codimension 2n of the orthogonal ones.

Line Hermitian Grassmann codes and their parameters

“…In [3], we started investigating some projective codes arising from subgeometries of the Grassmann variety associated to orthogonal and symplectic k-Grassmannians. We called such codes respectively orthogonal [3,5,6,7] and symplectic Grassman codes [4,6]. In the cases of line orthogonal and symplectic Grassmann codes, i.e.…”

“…In the cases of line orthogonal and symplectic Grassmann codes, i.e. for k = 2, we determined all the parameters; see [5], [7] and [4]. For both these families we also proposed in [6] an efficient encoding algorithm, based on the techniques of enumerative coding introduced in [12].…”

Enumerative coding for line polar Grassmannians with applications to codes

“…We summarize in the following theorems what is currently known about the parameters of these codes. [4], [5]). The known parameters [N, K, d] of P n,k := C(∆ n,k ) are…”

“…These codes have been introduced respectively in [2] and in [3] as linear codes arising from the Plücker embedding of polar Grassmannians of orthogonal or symplectic type. Some bounds on their minimum distance have been obtained: it has been proved in [4] for q odd and in [5] for q even that if k = 2 then the minimum distance of a line orthogonal Grassman code is q 4n−5 − q 3n−4 ; the minimum distance of a line symplectic Grassman code is q 4n−5 − q 2n−3 (see [3]). We shall set the notation in Section 1.1 and recall some notions about polar Grassmann codes in Section 1.2.…”

Implementing line-Hermitian Grassmann codes

“…In the present paper we shall be concerned with point enumerators of line Hermitian Grassmannians. In this way, we continue a project started in [4] where we introduced point enumerators for line polar Grassmannians of orthogonal [7,5] and symplectic type [3].…”