Hyperplane Sections of Grassmannians and the Number of MDS Linear Codes

@article{Ghorpade2001HyperplaneSO,
  title={Hyperplane Sections of Grassmannians and the Number of MDS Linear Codes},
  author={Sudhir R. Ghorpade and Gilles Lachaud},
  journal={Finite Fields and Their Applications},
  year={2001},
  volume={7},
  pages={468-506}
}
  • S. Ghorpade, G. Lachaud
  • Published 1 October 2001
  • Mathematics, Computer Science
  • Finite Fields and Their Applications
We obtain some effective lower and upper bounds for the number of (n,k)-MDS linear codes over Fq. As a consequence, one obtains an asymptotic formula for this number. These results also apply for the number of inequivalent representations over Fq of the uniform matroid or, alternately, the number of Fq-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… 
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Using a combinatorial approach to studying the hyperplane sections of Grassmannians, a bound is obtained on the number of higher dimensional subcodes of the Grassmann code having the minimum Hamming norm.
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