Finite Fields and Their Applications

S. R. Ghorpade and G. Lachaud MDS Codes and Grassmannians

Department of Mathematics, Indian Institute of Technology, Bombay,
Powai, Mumbai 400076, India

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Équipe ``Arithmétique et Théorie de l'Information'', Institut de Mathématiques
de Luminy, Luminy Case 907, 13288 Marseille, Cedex 9, France

Abstract:

We obtain some effective lower and upper bounds for the number of (n,k)-MDS linear codes over $\mathop{\mathbb{F} _{q}}\nolimits$. As a consequence, one obtains an asymptotic formula for this number. These results also apply for the number of inequivalent representations over $\mathop{\mathbb{F} _{q}}\nolimits$ of the uniform matroid, or, alternately, the number of $\mathop{\mathbb{F} _{q}}\nolimits$-rational points of certain open strata of Grassmannians. The techniques used in the determination of bounds for the number of MDS codes are applied to deduce several geometric properties of certain sections of Grassmannians by coordinate hyperplanes.