Minimum distance and the minimum weight codewords of Schubert codes

Abstract

Abstract. We consider linear codes associated to Schubert varieties in Grassmannians. A formula for the minimum distance of these codes was conjectured in 2000 and after having been established in various special cases, it was proved in 2008 by Xiang. We give an alternative proof of this formula. Further, we propose a characterization of the minimum weight codewords of Schubert codes by introducing the notion of Schubert decomposable elements of certain exterior powers. It is shown that codewords corresponding to Schubert decomposable elements are of minimum weight and also that the converse is true in many cases. A lower bound, and in some cases, an exact formula, for the number of minimum weight codewords of Schubert codes is also given. From a geometric point of view, these results correspond to determining the maximum number of Fq-rational points that can lie on a hyperplane section of a Schubert variety in a Grassmannian with its nondegenerate embedding in a projective subspace of the Plücker projective space, and also the number of hyperplanes for which the maximum is attained.

Hyperplane Sections of Determinantal Varieties over Finite Fields and Linear Codes

“…Adding all contributions to w r (t; ℓ, m) from Equations ( 8),( 9),( 10),( 11), (12), and ( 13), the theorem follows. ✷…”

“…We also show that the determinantal codes have a very low dual minimum distance (viz., 3), which makes them rather interesting from the point of view of coding theory. Analogous problems have been considered for other classical projective varieties such as Grassmannians, Schubert varieties, etc., leading to interesting classes of linear codes which have been of some current interest; see, for example, [17], [13], [14], [20], [12], and the survey [16].…”

Majority Logic Decoding for Certain Schubert Codes Using Lines in Schubert Varieties

“…The MDC was first proved, in generality, by Xiang [25]. A different proof of the MDC was given, and an attempt to give a classification of the minimum weight codewords was made in [10]. In the case, ℓ = 2, the weight distribution of the Schubert code C α (2, m) is known [20].…”

The weight spectrum of certain affine Grassmann codes

“…In the case when = 2, codewords of the Grassmann code C(2, m) are in one-to-one correspondence with the space of skew-symmetric matrices of dimension m × m. Nogin [11] used the classification of skew-symmetric matrices to give the weight spectrum of the Grassmann code C (2, m). In later work, Nogin [12] and Kaipa-Pillai [9] computed the weight spectrum of the Grassmann codes C (3,6) and C(3, 7) respectively. In general, the weight spectrum of Grassmann codes is not known.…”

“…This conjecture is known as the Minimum Distance Conjecture or the MDC. The MDC was proved, first by H. Chen [3] and Guerra-Vincenti [8] when = 2, then by Ghorpade-Tsfasman [7] for the Schubert divisor and finally by Xiang [19] and [6] for general Schubert codes. It is now well known that the Schubert code…”