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… 
Schubert varieties, linear codes and enumerative combinatorics
Higher Weights of Grassmann Codes
TLDR
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.
Codes on Linear Sections of Grassmannians
We study algebraic geometry linear codes defined by linear sections of the Grassmannian variety as codes associated to FFN$(1,q)$-projective varieties. As a consequence, we show that Schubert,
On Lagrangian–Grassmannian codes
TLDR
A new class of linear codes generated by this variety of smooth algebraic variety n(n + 1)/2 is constructed, the Lagrangian–Grassmannian codes.
Subclose Families, Threshold Graphs, and the Weight Hierarchy of Grassmann and Schubert Codes
TLDR
A number of properties of subclose families are proved, and its connection with the notion of threshold graphs and graphs with maximum sum of squares of vertex degrees is outlined.
On codes over FFN(1, q)-projective varieties
TLDR
Using these subvarieties, a unique subvariety that satisfies the Finite Field Nullstellensatz property is shown, for homogeneous linear polynomials over a finite field.
Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields
We consider the problem of determining the maximum number of common zeros in a projective space over a finite field for a system of linearly independent multivariate homogeneous polynomials defined
Enumerative Coding for Line Polar Grassmannians.
TLDR
The aim of the present paper is to present an efficient enumerative coding and decoding strategy for line polar Grassmann codes.
Codes on linear sections of the Grassmannian
We show that for every linear section of a Grassmannian over a finite field $${{\mathbb {F}}}_q$$ , in addition to the evaluation code associated to the $${{\mathbb {F}}}_q$$ -rational points of
...
...

References

SHOWING 1-10 OF 81 REFERENCES
Higher Weights of Grassmann Codes
TLDR
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.
Arcs in PG(n, q), MDS-codes and three fundamental problems of B. Segre — Some extensions
To each arc of PG(n, q) an algebraic hypersurface is associated. Using this tool new results on complete arcs are obtained. Since arcs and linear MDS-codes are equivalent objects, these results can
Formula for the number of [9, 3] MDS codes
We compute the number of MDS codes of length 9 and dimension 3 over all finite fields, or, what is essentially equivalent, the number of 9-arcs in the projective plane over a finite field.
The weight hierarchy of higher dimensional Hermitian codes
TLDR
The weight hierarchy, also known as the set of generalized Hamming weights, of the code is calculated and the higher weight distribution is also found.
THE NUMBER OF SUBSPACES OF A VECTOR SPACE.
Abstract : The number G sub n of subspaces of an n-dimensional vector space over GF(q) is studied by the symbolic calculus. This calculus provides a general technique for proving theorems in
Geometric approach to higher weights
The notion of higher (or generalized) weights of codes is just as natural as that of the classical Hamming weight. The authors adopt the geometric point of view and always treat the q-ary case. Some
Multiplicities of points on a Schubert variety in a minuscule GP
Schubert varieties and Demazure's character formula
In [3] M. Demazure constructed the so-called Bott-Samelson scheme (see also [6]) which gives desingularizations of the Schubert varieties in the flag manifold G/B. Here G denotes a semi-simple
General Galois geometries
This book is the second edition of the third and last volume of a treatise on projective spaces over a finite field, also known as Galois geometries. This volume completes the trilogy comprised of
Coherent cohomology on schubert subschemes of flag schemes and applications
...
...