Orbit Structure of Grassmannian G2,m and a Decoder for Grassmann Code C(2, m)

Abstract

In this article, we consider decoding Grassmann codes, linear codes associated to the Grassmannian and its embedding in a projective space. We look at the orbit structure of Grassmannian arising from the multiplicative group ${\mathbb {F}}_{q^{m}}^{*}$ in $GL_{m}(q)$ . We project the corresponding Grassmann code onto these orbits to obtain a subcode of a $q$ –ary Reed-Solomon code. We prove that some of these projections contain an information set of the parent Grassmann code. By improving the decoding capacity 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.