Affine Grassmann Codes

Duals of Affine Grassmann Codes and their Relatives


Peter Beelen

Department of Mathematics,
Technical University of Denmark,
DK 2800, Lyngby, Denmark

E-Mail: p.beelen@mat.dtu.dk

Sudhir R. Ghorpade

Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai 400076, India

E-Mail: srg@math.iitb.ac.in

and

Tom Høholdt

Department of Mathematics,
Technical University of Denmark,
DK 2800, Lyngby, Denmark

E-Mail: T.Hoeholdt@mat.dtu.dk


Abstract

Affine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work [2]. Here we consider, more generally, affine Grassmann codes of a given level. We explicitly determine the dual of an affine Grassmann code of any level and compute its minimum distance. Further, we ameliorate the results of [2] concerning the automorphism group of affine Grassmann codes. Finally, we prove that affine Grassmann codes and their duals have the property that they are linear codes generated by their minimum-weight codewords. This provides a clean analogue of a corresponding result for generalized Reed-Muller codes.


1 Introduction 1
2 Preliminaries 2
3 Minimum Distance 5
4 Automorphisms 6
5 Duality 8
6 Generation by Minimum Weight Codewords 13
References 20


This paper is publihsed in the IEEE Transactions on Information Theory, Vol. 58, No. 6 (2012), pp. 3843-3855. [Preprint Version: arXiv.math/1107.3438v1 (July 2011)]

Download the full paper as:

PDF File Postscript File DVI File.


Back to the List of Publications

Back to the Sudhir Ghorpade's Home Page