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Codes associated to Schubert varieties in G(2,5) over F_2

Mahesh S. Wavare
2019 New Trends in Mathematical Sciences  
In this paper, we consider the problem of determining generalized spectrum of linear codes associated to Schubert subvarieties of Grassmannians.  ...  A linear [n, k] q -code is a k-dimensional subspace of the n-dimensional vector space F n q .  ...  Acknowledgement The auther is very much thankful to UGC for giving financial assitant for Minor Research Project under XII plan and File No. 47-475/12(WRO)dated 13-03-2013.  ... 
doi:10.20852/ntmsci.2019.343 fatcat:cypwepmownfzfld5lc2wilv43y

The structure of dual Schubert union codes [article]

Fernando L. Piñero
2017 arXiv   pre-print
In this article we prove that Schubert union codes are Tanner codes constructed with the point--line incidence geometry that Schubert varieties inherit from the Grassmannian.  ...  We find sufficient conditions on the initial set and the initial positions such that a codeword is determined from the component codes only.  ...  of Algebraic Geometry in Coding Theory and Cryptography.  ... 
arXiv:1410.3703v6 fatcat:e7yfdoatkbburfeahs4evyumae

Schubert unions in Grassmann varieties

Johan P. Hansen, Trygve Johnsen, Kristian Ranestad
2007 Finite Fields and Their Applications  
We study unions of Schubert cycles of Grassmann varieties G(l, m) over a field F . We compute their linear span and, in positive characteristic, their number of F q -rational points.  ...  We discuss the maximum number of F q -rational points for Schubert unions of a given spanning dimension, and as an application to coding theory, we study the parameters and support weights of the well-known  ...  The second author was supported in part by the Norwegian Research Council and thanks for this support and for the kind hospitality extended by Aarhus University, Denmark, where the support was spent.  ... 
doi:10.1016/j.ffa.2007.06.003 fatcat:fwen7x3xcnhf3d62g5hslj535q

Orbit Structure of Grassmannian G_2, m and a decoder for Grassmann code C(2, m) [article]

Fernando Piñero, Prasant Singh
2021 arXiv   pre-print
We look at the orbit structure of Grassmannian arising from the natural action of multiplicative group of certain finite field extension.  ...  By improving the efficiency of Peterson's decoding algorithm for the projected subcodes, we prove that one can correct up to ⌊ d-1/2⌋ errors for Grassmann code, where d is the minimum distance of Grassmann  ...  Schubert codes, linear codes associated to Schubert varieties in Grassmannian are one among these classes.  ... 
arXiv:2008.12067v2 fatcat:ugceddd67fa2pm5y2ew6xvrwvm

PD-sets for binary RM-codes and the codes related to the Klein quadric and to the Schubert variety of PG(5,2)

Hans-Joachim Kroll, Rita Vincenti
2008 Discrete Mathematics  
We find PD-sets for some binary Grassmann codes, that is, for the projective Reed-Muller codesR(1, 4) andR(1, 5) and for the codes related to the Klein quadric KQ of PG(5, 2) and to the Schubert subvariety  ...  We consider the following simplexes of PG(4, 2): For every simplex A there are 5! collineations mapping A to A 0 . Choose for i ∈ {1, . . . , 31} one collineation i such that i (A i ) = A 0 .  ...  Codes arising from varieties are of particular interest. As shown in [10] , if X is a variety having an automorphism group A, then A induces a linear automorphism group of the related code C.  ... 
doi:10.1016/j.disc.2006.11.057 fatcat:oktn27eyejhgzjhjh375hpxvmi

Numerical Schubert Calculus in Macaulay2 [article]

Anton Leykin, Abraham Martín del Campo, Frank Sottile, Ravi Vakil, Jan Verschelde
2021 arXiv   pre-print
The Macaulay2 package NumericalSchubertCalculus provides methods for the numerical computation of Schubert problems on Grassmannians.  ...  One is in the scripting language of Macaulay2 using the package NumericalAlgebraicGeometry, and the other is in the compiled code of PHCpack.  ...  The authors thank the American Institute of Mathematics for supporting the project through their SQuaREs program. The package NumericalSchubertCalculus is included in Macaulay2 version 1.18.  ... 
arXiv:2105.04494v1 fatcat:i4ioapn5rnfoflvovg35k22c5e

Rational curves on Calabi-Yau manifolds: verifying predictions of Mirror Symmetry [article]

Sheldon Katz
1993 arXiv   pre-print
This increases the number of verified predictions from 3 to 65. Calculations are performed using the Maple package schubert written by Katz and Strømme.  ...  In this paper, the numbers of lines and conics on various hypersurfaces which satisfy certain incidence properties are calculated, and shown to agree with the numbers predicted by Greene, Morrison, and  ...  The short schubert code is not included here, but is available upon request.  ... 
arXiv:alg-geom/9301006v2 fatcat:67wwbn5qz5btxbwzqowytjqhhq

Subspace Codes for Random Networks Based on Plücker Coordinates and Schubert Cells [article]

Anirban Ghatak
2013 arXiv   pre-print
The selection of these Schubert cells is based on the subset distance of tuples corresponding to the Plücker coordinate matrices associated with the subspaces contained in the respective Schubert cells  ...  These codes may be described as the union of constant dimension subspace codes restricted to selected Schubert cells.  ...  ACKNOWLEDGMENT The author is grateful to Dilip P. Patil, Sumanta Mukherjee and Smarajit Das for their assistance.  ... 
arXiv:1301.6362v2 fatcat:wx6w6hwzendwfmzetik3didxoe

Page 4408 of Mathematical Reviews Vol. , Issue 95g [page]

1995 Mathematical Reviews  
Reformulating these problems in the setting of Schubert varieties unifies the separate conditions arising from pole placement and from pole-zero cancellation into a single condition: (*) Given f points  ...  We show that these algorithms induce linear dynamical systems on various manifolds of rational functions. Applications to coding theory are considered.”  ... 

Experimentation in the Schubert Calculus [article]

Abraham Martin del Campo, Frank Sottile
2013 arXiv   pre-print
Many aspects of Schubert calculus are easily modeled on a computer. This enables large-scale experimentation to investigate subtle and ill-understood phenomena in the Schubert calculus.  ...  A similarly rich story concerning intrinsic structure, or Galois groups, of Schubert problems is also beginning to emerge from experimentation.  ...  We close with the observation that when we computed real osculating instances of this Schubert problem (as part of the experiment described in Section 4), we only found either zero or four real solutions  ... 
arXiv:1308.3284v1 fatcat:tqqsoxvmxncjfftnusrnug3tfe

The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials [article]

Philippe Nadeau, Vasu Tewari
2021 arXiv   pre-print
We compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes.  ...  It is based in part on a relation between the numbers a_w and principal specializations of Schubert polynomials.  ...  Acknowledgments We are grateful to Sara Billey, Sean Griffin, and particularly Alex Woo for numerous helpful discussions at the early stages of this project.  ... 
arXiv:2005.12194v2 fatcat:kadi7ray4vftfpjpzrr4jpmdqa

On the EKL-Degree of a Weyl Cover [article]

Joseph Knight, Ashvin Swaminathan, Dennis Tseng
2020 arXiv   pre-print
More than four decades ago, Eisenbud, Khimšiašvili, and Levine introduced an analogue in the algebro-geometric setting of the notion of local degree from differential topology.  ...  We use knowledge of the cohomology ring of partial flag varieties as a key input in our proofs, and our computations give interesting explicit examples in the field of 𝔸^1-enumerative geometry.  ...  The first author was supported by Purdue University, and the second and third authors were supported by NSF Graduate Research Fellowship Grants 1656466 and 1745303.  ... 
arXiv:1907.03856v2 fatcat:owrbk3z74zeljm6rijd6wnc7yq

The family G_T of graded quotients of k[x,y] of given Hilbert function [article]

A. Iarrobino, J. Yameogo (Universite de Nice)
2004 arXiv   pre-print
The dimension of the cell V(E) is the number of difference one hooks of P(E).  ...  A main tool is a combinatorial "hook code" for the partition P(E), that gives the image of the cell V(E) in SGrass(T).  ...  . , y j be the subspace of degree-j homogeneous polynomials. In Section 2-A we first describe how a d-dimensional vector space V of R j is a linear system on the projective lineP 1 .  ... 
arXiv:alg-geom/9709021v2 fatcat:w6doisvw3zbzhfpuwxlq6zhbu4

The many faces of modern combinatorics [article]

Cristian Lenart
2015 arXiv   pre-print
The goal is to give a big picture of its many interactions with other areas of mathematics, such as: group theory, representation theory, commutative algebra, geometry (including algebraic geometry), topology  ...  A representation of a group G is a group homomorphism from G to the group of invertible linear maps from a certain vector space to itself (i.e., the general linear group of the vector space).  ...  We now turn to the problem of the singular locus of Schubert varieties in SL n /B.  ... 
arXiv:1503.04240v1 fatcat:vlzdvhv2azfx7oxop3iklccrua

Staircase diagrams and the enumeration of smooth Schubert varieties

Edward Richmond, William Slofstra
2020 Discrete Mathematics & Theoretical Computer Science  
In particular, we show that rationally smooth Schubert varieties are in bijection with a new combinatorial data structure called staircase diagrams.  ...  International audience In this extended abstract, we give a complete description and enumeration of smooth and rationally smooth Schubert varieties in finite type.  ...  Acknowledgements The authors thank Sara Billey and Jim Carrell for helpful conversations, and Sara Billey for providing data on the number of smooth and rationally smooth Schubert varieties.  ... 
doi:10.46298/dmtcs.6411 fatcat:gm7efqco7vaazapnau7jlp2nmu
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