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)]
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