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

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.

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Acknowledgments

The authors wish to thank Antonia Wachter-Zeh for many helpful discussions. They also thank the anonymous reviewers for their valuable comments and suggestions that helped to improve the presentation of the paper. J. Rosenthal was partially supported by Swiss National Science Foundation Grant Nos. 138080 and 149716. N. Silberstein was supported in part at the Technion by a Fine Fellowship. A.-L. Trautmann was partially supported by Forschungskredit of the University of Zurich, Grant No. 57104103, and Swiss National Science Foundation Fellowship No. 147304. Parts of this work were presented at the International Workshop on Coding and Cryptography 2013 in Bergen, Norway, and appear in its proceedings [30].

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Correspondence to Anna-Lena Trautmann.

Additional information

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Rosenthal, J., Silberstein, N. & Trautmann, A. On the geometry of balls in the Grassmannian and list decoding of lifted Gabidulin codes. Des. Codes Cryptogr. 73, 393–416 (2014). https://doi.org/10.1007/s10623-014-9932-x

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Keywords

  • Grassmannian
  • Projective space
  • Subspace codes
  • Network coding
  • List decoding

Mathematics Subject Classification

  • 11T71
  • 14G50