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Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 817–830 | Cite as

The weight spectrum of certain affine Grassmann codes

  • Fernando PiñeroEmail author
  • Prasant Singh
Article
  • 63 Downloads
Part of the following topical collections:
  1. Special Issue: Finite Geometries

Abstract

We consider the linear code corresponding to a special affine part of the Grassmannian \({G_{2,m}}\), which we denote by \({C^{\mathcal {A}}(2, m)}\). This affine part is the complement of the Schubert divisor of \({G_{2,m}}\). In view of this, we show that there is a projection of Grassmann code onto the affine Grassmann code which is also a linear isomorphism. This implies that the dimensions of Grassmann codes and affine Grassmann codes are equal. The projection gives a 1–1 correspondence between codewords of Grassmann codes and affine Grassmann codes. Using this isomorphism and the correspondence between codewords, we give a skew–symmetric matrix in some standard block form corresponding to every codeword of \({C^{\mathcal {A}}(2, m)}\). The weight of a codeword is given in terms of the rank of some blocks of this form and it is shown that the weight of every codeword is divisible by some power of q. We also count the number of skew–symmetric matrices in the block form to compute the weight spectrum of the affine Grassmann code \({C^{\mathcal {A}}(2, m)}\).

Keywords

Grassmann varieties Schubert varieties Affine Grassmannian Linear codes Weight distribution 

Mathematics Subject Classification

14M15 51M35 94B27 

Notes

Acknowledgements

The first named author is very grateful for all the support received from the organizers of the fifth Irsee conference on Finite Geometries. The second named author would like to acknowledge H. C. Ørsted cofund postdoc fellowship for the project “Understanding Schubert Codes”. We are also grateful for the referees and their helpful comments which improved this article.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Puerto Rico at PoncePoncePuerto Rico
  2. 2.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKgs. LyngbyDenmark

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