The finite Grassmannian Gq(k, n) is defined as the set of all k -dimensional subspaces of the ambient space Fq n. Subsets of the finite Grassmannian are called constant dimension codes and have recently found an application in random network coding. In this setting codewords from G q(k, n) are sent through a network channel and, since errors may occur during transmission, the received words can possibly lie in G q(k, n), where k' ≠ k. In this paper, we study the balls in G q(k, n) with center that is not necessarily in Gq (k, n). We describe the balls with respect to two different metrics, namely the subspace and the injection metric. Moreover, we use two different techniques for describing these balls, one is the Plücker embedding of Gq(k, n), and the second one is a rational parametrization of the matrix representation of the codewords. With these results, we consider the problem of list decoding a certain family of constant dimension codes, called lifted Gabidulin codes. We describe a way of representing these codes by linear equations in either the matrix representation or a subset of the Plücker coordinates. The union of these equations and the linear and bilinear equations which arise from the description of the ball of a given radius provides an explicit description of the list of codewords with distance less than or equal to the given radius from the received word.

Rosenthal, Joachim; Silberstein, Natalia; Trautmann, Anna Lena (2014). *On the geometry of balls in the Grassmannian and list decoding of lifted Gabidulin codes.* Designs, Codes and Cryptography, 73(2):393-416.

## Abstract

The finite Grassmannian Gq(k, n) is defined as the set of all k -dimensional subspaces of the ambient space Fq n. Subsets of the finite Grassmannian are called constant dimension codes and have recently found an application in random network coding. In this setting codewords from G q(k, n) are sent through a network channel and, since errors may occur during transmission, the received words can possibly lie in G q(k, n), where k' ≠ k. In this paper, we study the balls in G q(k, n) with center that is not necessarily in Gq (k, n). We describe the balls with respect to two different metrics, namely the subspace and the injection metric. Moreover, we use two different techniques for describing these balls, one is the Plücker embedding of Gq(k, n), and the second one is a rational parametrization of the matrix representation of the codewords. With these results, we consider the problem of list decoding a certain family of constant dimension codes, called lifted Gabidulin codes. We describe a way of representing these codes by linear equations in either the matrix representation or a subset of the Plücker coordinates. The union of these equations and the linear and bilinear equations which arise from the description of the ball of a given radius provides an explicit description of the list of codewords with distance less than or equal to the given radius from the received word.

## Citations

## Altmetrics

## Downloads

## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 21 February 2014 |

Deposited On: | 30 Sep 2014 10:43 |

Last Modified: | 05 Apr 2016 18:22 |

Publisher: | Springer |

ISSN: | 0925-1022 |

Publisher DOI: | https://doi.org/10.1007/s10623-014-9932-x |

Related URLs: | http://www.scopus.com/redirect/linking.url?targetURL=http%3a%2f%2fdx.doi.org%2f10.1007%2fs10623-014-9932-x&locationID=1&categoryID=4&eid=2-s2.0-84905238564&issn=09251022&linkType=ViewAtPublisher&year=2014&origin=recordpage&dig=731394ce1f05c3d7d05f1c9d77e1 (Publisher) |

## Download

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.

You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.