Designs, Codes and Cryptography

, Volume 73, Issue 2, pp 393–416 | Cite as

On the geometry of balls in the Grassmannian and list decoding of lifted Gabidulin codes

  • Joachim Rosenthal
  • Natalia Silberstein
  • Anna-Lena Trautmann
Article

Abstract

The finite Grassmannian \(\mathcal {G}_{q}(k,n)\) is defined as the set of all \(k\)-dimensional subspaces of the ambient space \(\mathbb {F}_{q}^{n}\). 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 \(\mathcal {G}_{q}(k,n)\) are sent through a network channel and, since errors may occur during transmission, the received words can possibly lie in \(\mathcal {G}_{q}(k',n)\), where \(k'\ne k\). In this paper, we study the balls in \(\mathcal {G}_{q}(k,n)\) with center that is not necessarily in \(\mathcal {G}_{q}(k,n)\). 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 \(\mathcal {G}_{q}(k,n)\), 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.

Keywords

Grassmannian Projective space Subspace codes Network coding  List decoding 

Mathematics Subject Classification

11T71 14G50 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Joachim Rosenthal
    • 1
  • Natalia Silberstein
    • 2
  • Anna-Lena Trautmann
    • 3
  1. 1.Institute of MathematicsUniversity of ZurichZurichSwitzerland
  2. 2.Department of Computer ScienceTechnion—Israel Institute of TechnologyHaifaIsrael
  3. 3.Department of Electrical and Electronic EngineeringUniversity of MelbourneParkvilleAustralia

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