Hyperplane sections of Grassmannians and the number of MDS linear codes
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 \(q\).
scientific article; zbMATH DE number 1763654
scientific article