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

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 Grass...

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Bibliographic Details
Published in:arXiv.org 2021-06
Main Authors: Piñero, Fernando, Singh, Prasant
Format: Article
Language:English
Subjects:
Algorithms
Codes
Decoding
Fields (mathematics)
Online Access:Get full text
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Summary: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.
ISSN:2331-8422