Towards the characterization of minimum weight codewords of Schubert codes

TL;DR

This paper proves a conjecture about the minimum weight codewords of Schubert codes, showing they are given by Schubert decomposable codewords for most values of q, advancing understanding of these algebraic codes.

Contribution

It confirms the conjecture that minimum weight codewords are Schubert decomposable for all but finitely many q, extending previous proofs and conjectures.

Findings

Confirmed the conjecture for all but finitely many q

Established the validity of the minimum weight codeword characterization

Extended previous proofs to a broader class of Schubert varieties

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 $\Omega_{\a}(\ell, V_m)$ for all but finitely many values of $q$.