Designs, Codes and Cryptography

, Volume 86, Issue 8, pp 1767–1792 | Cite as

Rank metric codes and zeta functions

  • I. Blanco-Chacón
  • E. Byrne
  • I. Duursma
  • J. Sheekey


We define the rank metric zeta function of a code as a generating function of its normalized q-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank metric codes. We further prove a functional equation and derive an upper bound for the minimum distance in terms of the reciprocal roots of the zeta function. Finally, we show invariance under suitable puncturing and shortening operators and study the distribution of zeroes of the zeta function for a family of codes.


Rank metric code Zeta function Weight enumerator Maximum-rank-distance Binomial moments Gaussian binomial coefficient 

Mathematics Subject Classification

11T71 94B05 94B27 94B60 94B65 94B99 



Blanco-Chacón was partially supported by the Science Foundation of Ireland (SFI 13/IA/1914), and the Spanish National Research Council (MTM2013-42135P). Duursma was partially supported by NSF (CCF-1619189).


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity College DublinDublinIreland
  2. 2.Department of MathematicsUniversity of Illinois Urbana-ChampaignChampaignUSA

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