Line Hermitian Grassmann codes and their parameters

Abstract

In this paper we introduce and study line Hermitian Grassmann codes as those subcodes of the Grassmann codes associated to the 2-Grassmannian of a Hermitian polar space defined over a finite field. In particular, we determine the parameters and characterize the words of minimum weight.

Minimum distance of orthogonal line-Grassmann codes in even characteristic

“…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].…”

“…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.…”

Implementing line-Hermitian Grassmann codes

“…It t = 0 then ψ(D 0 ) = ψ E (∅) is clearly the number of lines of H m .If t > 0 system (6) can be written as…”

“…In a series of papers we have investigated codes arising when Ω is a proper subvariety of G(m, k), namely when Ω is the image under the Plücker embedding of a polar line Grassmannian; see [2,7,5] for the orthogonal case and [3] for the symplectic case. Following the same approach as of [2], we defined in [6] Hermitian Grassmann codes as those projective codes arising from the Plücker embedding of a Hermitian Grassmannian (see Equation ( 1)) and we determined their minimum distance, also characterizing the words of minimum weight.…”

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In [6] we introduced line Hermitian Grassmann codes and determined their parameters. The aim of this paper is to present (in the spirit of [4]) 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.

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