Hyperplane sections of Grassmannians and the number of MDS linear codes
type: Item
instance of: scholarly article
title: Hyperplane sections of Grassmannians and the number of MDS linear codes
zbMATH Open document ID: 1007.94024
author: Sudhir R. Ghorpade
author: Gilles Lachaud
published in: Finite Fields and their Applications
publication date: 2002-07-04T00:00:00Z
review text: The authors obtain upper and lower bounds for the number of \((n,k)\)-MDS linear codes over \({\mathbb F}_q\), and an asymptotic formula for this number. \textit{A. N. Skorobogatov} [Linear codes, strata of Grassmannians, and the problems of Segre. In: Coding theory and algebraic geometry, Proc. Int. Workshop, Luminy/Fr. 1991, Lect. Notes Math. 1518, 210-223 (1992; Zbl 0809.94022)] showed that this number is related to the number of \({\mathbb F}_q\)-rational points of the Grassmannian of all \(k\)-dimensional linear subspaces of \({\mathbb F}_q^n\) for which all the Plücker coordinates are nonzero. He also showed that this number is the number of inequivalent representations of the uniform matroid \(U_{k,n}\) over \({\mathbb F}_q\), and it is related to three classical questions of B. Segre on \(n\)-arcs and the rational normal curve in \({\mathbb P}^{k-1}\). The authors obtain bounds on the number of rational points on hyperplane sections of the Grassmannian by using results on the parameters of Grassmann codes obtained by \textit{D. Yu. Nogin} [Codes associated to Grassmannians. In: Pellikaan, R. (ed.) et al., Arithmetic, geometry, and coding theory. Proc. Int. Conf. CIRM, Luminy, France, 1993, Walter de Gruyter, 145-154 (1996; Zbl 0865.94032)]. The authors also obtain results on geometric properties of these hyperplane sections of the Grassmannian. In the final section, they give some tables containing numerical values for the lower and upper bounds on the number of MDS codes for certain small values of \(n, k\), and \(q\).
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 1763654
zbMATH Keywords: Grassmannian
zbMATH Keywords: MDS code
zbMATH Keywords: Grassmann code
zbMATH Keywords: hyperplane section
zbMATH Keywords: uniform matroid
zbMATH Keywords: arcs in projective space
reviewed by: Robert F. Lax
MaRDI profile type: Publication
OpenAlex ID: W1984633059
cites work: Q4039722
cites work: Schubert varieties and Demazure's character formula
cites work: Q4146776
cites work: Q3726125
cites work: Arcs in PG(n,q), MDS-codes and three fundamental problems of B. Segre - some extensions
cites work: On M.D.S. codes, arcs in PG(n,q) with q even, and a solution of three fundamental problems of B. Segre
cites work: Determinantal rings
cites work: Ladder determinantal rings
cites work: Q4317713
cites work: Young multitableaux and higher dimensional determinants
cites work: Q4485866
cites work: Rings of geometries. II
cites work: Q5659439
cites work: Q5592270
cites work: Gröbner bases and multiplicity of determinantal and Pfaffian ideals
cites work: Q4003762
cites work: The weight hierarchy of higher dimensional Hermitian codes
cites work: Grassmannians and their Schubert subvarieties are arithmetically Cohen- Macaulay
cites work: Q5814388
cites work: Formula for the number of [9,3] MDS codes
cites work: Schubert Calculus
cites work: Q4336971
cites work: Multiplicities of points on a Schubert variety in a minuscule G/P
cites work: The arithmetic Cohen-Macaulay character of Schubert schemes
cites work: Q4146667
cites work: Q5611879
cites work: Normality of Schubert Varieties
cites work: Q3871006
cites work: Q3721319
cites work: Q4402068
cites work: Q4336974
cites work: Q3686628
cites work: Projective normality of flag varieties and Schubert varieties
cites work: The number of MDS [7,3] codes on finite fields of characteristic 2
cites work: Q4331868
cites work: Q4206872
cites work: Q4027652
cites work: On the number of representations of matroids over finite fields
cites work: Coherent cohomology on Schubert subschemes of flag schemes and applications
cites work: Q3997777
cites work: Geometric approach to higher weights
cites work: Generalized Hamming weights for linear codes
DOI: 10.1006/FFTA.2000.0299
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description: scientific article; zbMATH DE number 1763654