Abstract
In analogy with the Singleton defect for classical codes, we propose a definition of rank defect for rank-metric codes. We characterize codes whose rank defect and dual rank defect are both zero, and prove that the rank distribution of such codes is determined by their parameters. This extends a result by Delsarte on the rank distribution of MRD codes. In the general case of codes of positive defect, we show that the rank distribution is determined by the parameters of the code, together with the number of codewords of small rank. Moreover, we prove that if the rank defect of a code and its dual are both one, and the dimension satisfies a divisibility condition, then the number of minimum-rank codewords and dual minimum-rank codewords is the same. Finally, we discuss how our results specialize to Fqm-linear rank-metric codes in vector representation.
| Original language | English |
|---|---|
| Pages (from-to) | 1-16 |
| Number of pages | 16 |
| Journal | Designs, Codes and Cryptography |
| Volume | 86 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 1 Jan 2018 |
Funding
Acknowledgements The authors wish to thank the anonymous referees for suggestions that improved the legibility of the paper. We are grateful to one of the referees for suggesting to look for a generalization of Theorem 27 along the lines of [6, Theorem 7.3.1], which lead to Theorem 30. J. de la Cruz was partially supported by COLCIENCIAS through Project No. 121571250178. Part of the work was done while J. de la Cruz was visiting the University of Zurich. The first author thanks Joachim Rosenthal for the invitation. E. Gorla and A. Ravagnani were partially supported by the Swiss National Science Foundation through Grant No. 200021_150207 and by the ESF COST Action IC1104. H. López was partially supported by CONACyT and by the Swiss Confederation through the Swiss Government Excellence Scholarship No. 2014.0432.
Keywords
- MRD, QMRD, and dually-QMRD codes
- Rank distribution
- Rank-metric codes