Decomposable Subspaces, Linear Sections of Grassmann Varieties, and Higher Weights of Grassmann Codes
# Decomposable Subspaces, Linear Sections of Grassmann Varieties, and Higher Weights of Grassmann Codes

### Sudhir R. Ghorpade

#### *
Department of Mathematics *

Indian Institute of Technology Bombay

Powai, Mumbai 400076, India

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

###
Arunkumar R. Patil

#### *
Department of Mathematics *

Indian Institute of Technology Bombay

Powai, Mumbai 400076, India

E-Mail: arun.iitb@gmail.com

### and

### Harish K. Pillai

#### *
Department of Electrical Engineering*

Indian Institute of Technology Bombay

Powai, Mumbai 400076, India

E-Mail: hp@ee.iitb.ac.in

## Abstract

Given a homogeneous component of an exterior algebra, we characterize those subspaces in which every nonzero element is decomposable. In geometric terms, this corresponds to characterizing the projective linear subvarieties of the Grassmann variety with its Plucker embedding. When the base field is finite, we consider the more general question of determining the maximum number of points on sections of Grassmannians by linear subvarieties of a fixed (co)dimension. This corresponds to a known open problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties. We recover most of the known results as well as prove some new results. In the process we obtain, and utilize, a simple generalization of the Griesmer-Wei bound for arbitrary linear codes.

1 | Introduction | 1 |

2 | Decomposable Subspaces | 3 |

3 | Duality and the Hodge Star Operator | 7 |

4 | Griesmer-Wei Bound and its Generalization | 9 |

5 | The Grassman Code *C(l,m)* | 10 |

6 | Higher Weights of the Grassman Code *C(2,m)* | 12 |

| References | 16 |

This paper is published in *Finite Fields and
their Applications*, Vol. 15, No. 1 (2009), 54-68.

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