Enumeration of Splitting Subspaces over Finite Fields
# Enumeration of Splitting Subspaces over Finite Fields

### Sudhir R. Ghorpade

#### *
Department of Mathematics *

Indian Institute of Technology Bombay

Powai, Mumbai 400076, India

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

## and

### Samrith Ram

#### *
Department of Mathematics *

Indian Institute of Technology Bombay

Powai, Mumbai 400076, India

E-Mail: samrith@gmail.com

## Abstract

We discuss an elementary, yet unsolved, problem of Niederreiter concerning the
enumeration of a class of subspaces of finite dimensional vector spaces over
finite fields. A short and self-contained account of some recent progress on
this problem is included and some related problems are discussed.

1 | Introduction | 1 |

2 | Easy cases and guesse | 2 |

3 | Splitting planes | 3 |

4 | Refinements and Extensions | 5 |

A |
Appendix: Vector Recurrences and Singer Cycles | 8 |

| References | 10 |

This paper is published in:
Arithmetic, Geometry, Cryptography and Coding Theory,
(*Luminy, France, March 2011*
),
Y. Aubry, C. Ritzenthaler, and A. Zykin Eds.,
*Contemporary Mathematics*, Vol. 574,
American Mathematical Society, Providence, RI, 2012, pp. 49-58.
[Preprint Version:
arXiv.math/1203.1849 (Mar 2012)]

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