Orbit Structure of Grassmannian $G_{2, m}$ and a decoder for Grassmann code $C(2, m)$

TL;DR

This paper studies the orbit structure of Grassmannian $G_{2,m}$ and introduces a decoding method for Grassmann codes by projecting onto orbits, enabling correction of up to half the minimum distance.

Contribution

It presents a novel decoding approach for Grassmann codes using orbit projections and improved Peterson's decoding for subcodes, enhancing error correction capabilities.

Findings

Projected subcodes contain an information set of the Grassmann code.

The decoding method corrects up to -1/2 errors, where d is the minimum distance.

Orbit structure analysis leads to efficient decoding algorithms.

Abstract

In this manuscript, we consider decoding Grassmann codes, linear codes associated to Grassmannian of planes in an affine space. We look at the orbit structure of Grassmannian arising from the natural action of multiplicative group of certain finite field extension. We project the corresponding Grassmann code onto these orbits to obtain a few subcodes of certain Reed-Solomon code. We prove that some of these projected codes contains an information set of the parent Grassmann code. By improving the efficiency of Peterson's decoding algorithm for the projected subcodes, we prove that one can correct up to $\lfloor d-1/2\rfloor$ errors for Grassmann code, where $d$ is the minimum distance of Grassmann code.