A note on the weight spectrum of the Schubert code \(C_{\alpha }(2, m)\)

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Abstract

We consider the Schubert code \(C_{\alpha }(2, m)\) associated to the \(\mathbb {F}_q\)-rational points of the Schubert variety \(\Omega _{\alpha }(2,m)\) in the Grassmannian \(G_{2,m}\). A correspondence between codewords of \(C_{\alpha }(2, m)\) and skew-symmetric matrices of certain special form is given. Using this correspondence, we give a formula for all possible weights of codewords in \(C_{\alpha }(2, m)\). It is shown that the weight of each codeword is divisible by certain power of q. Further, a formula for the weight spectrum of the Schubert code \(C_{\alpha }(2, m)\) is given.

Keywords

Grassmann code Schubert code Weight spectrum Weight enumerator polynomial 

Mathematics Subject Classification

94B27 11T71 14M15 
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Notes

Acknowledgements

The authors would like to thank the anonymous referees for their suggestions to improve the article. We would also like to thank S.R. Ghorpade and A.R. Patil for their warm hospitality.

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

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Puerto Rico at PoncePonceUSA
  2. 2.Department of MathematicsIndian Institute of Technology BombayMumbaiIndia

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