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Towards the characterization of minimum weight codewords of Schubert codes

Mrinmoy Datta, Tiasa Dutta, Trygve Johnsen

Abstract

A conjectural formula for the minimum weight of Schubert codes was conjectured by Ghorpade in 2000. This was established by Xiang in 2008. In 2018, Ghorpade and Singh provided a new proof of this conjecture. Moreover, they also conjectured that the minimum weight codewords of the Schubert codes $C_\a (\ell, m)$ are given by the so-called \emph{Schubert decomposable codewords}. We prove the validity of the conjecture for all Schubert varieties $Ω_{\a}(\ell, V_m)$ for all but finitely many values of $q$.

Towards the characterization of minimum weight codewords of Schubert codes

Abstract

A conjectural formula for the minimum weight of Schubert codes was conjectured by Ghorpade in 2000. This was established by Xiang in 2008. In 2018, Ghorpade and Singh provided a new proof of this conjecture. Moreover, they also conjectured that the minimum weight codewords of the Schubert codes are given by the so-called \emph{Schubert decomposable codewords}. We prove the validity of the conjecture for all Schubert varieties for all but finitely many values of .
Paper Structure (10 sections, 12 theorems, 76 equations)

This paper contains 10 sections, 12 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Grassmannian, Plücker projective space, and Plücker embedding
  4. Schubert subvarieties of $G(\ell, V_m)$
  5. Hyperplanes, their decomposability, and Schubert decomposability
  6. Schubert varieties over finite fields: a decomposition into "smaller" Schubert varieties, and a combinatorial upper bound
  7. Enumeration of Schubert varieties over finite fields
  8. A combinatorial bound on Schubert subvarieties
  9. A decomposition of Schubert subvarieties and hyperplane sections
  10. Main Theorem

Key Result

Theorem 3.2

X, GS We have $e_{\alpha} (\ell, m) = |\Omega_\alpha (\ell, V_{m}, \mathcal{B})| - q^{\delta ({\alpha})}$.

Theorems & Definitions (39)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3: Schubert Decomposability, GS
  • Theorem 3.2
  • Conjecture 1: MWCC, GS
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 29 more