Geometric decoding of subspace codes with explicit Schubert calculus applied to spread codes

TL;DR

This paper introduces an improved geometric decoding algorithm for subspace codes, leveraging Schubert calculus and algebraic geometry, with applications to spread codes, enhancing efficiency and connecting to finite geometry.

Contribution

It presents a novel, more efficient decoding algorithm for geometric subspace codes using explicit Schubert calculus, applied to spread codes.

Findings

Complexity is substantially improved over previous algorithms.

Connections to finite geometry are established.

Applied successfully to Desarguesian spread codes.

Abstract

This article is about a decoding algorithm for error-correcting subspace codes. A version of this algorithm was previously described by Rosenthal, Silberstein and Trautmann. The decoding algorithm requires the code to be defined as the intersection of the Pl\"ucker embedding of the Grassmannian and an algebraic variety. We call such codes \emph{geometric subspace codes}. Complexity is substantially improved compared to the algorithm by Rosenthal, Silberstein and Trautmann and connections to finite geometry are given. The decoding algorithm is applied to Desarguesian spread codes, which are known to be defined as the intersection of the Pl\"ucker embedding of the Grassmannian with a linear space.