Maximum-rank distance (MRD) code[13] 

Description

Also called an optimal rank-distance code. An \([n\times m,k,d]_q\) rank-metric code whose parameters are such that the Singleton-like bound \begin{align} k \leq \max(n, m) (\min(n, m) - d + 1) \tag*{(1)}\end{align} become an equality.

Realizations

Useful for error and erasure correction in network coding [4,5].

Parent

Cousins

  • Maximum distance separable (MDS) code — MRD codes are matrix-code analogues of MDS codes.
  • Reed-Solomon (RS) code — MRD rank-metric codes can be thought of as matrix analogues of MDS Reed-Solomon codes as both constructions utilize a Vandermonde matrix [6].
  • Gabidulin code — Gabidulin codes over \(GF(q^N)\) with maximum rank-distance, when expressed as matrices over \(GF(q)\), are MRD codes.

References

[1]
P. Delsarte, “Bilinear forms over a finite field, with applications to coding theory”, Journal of Combinatorial Theory, Series A 25, 226 (1978) DOI
[2]
E. M. Gabidulin, "Theory of Codes with Maximum Rank Distance", Problemy Peredachi Informacii, Volume 21, Issue 1, 3–16 (1985)
[3]
R. M. Roth, “Maximum-rank array codes and their application to crisscross error correction”, IEEE Transactions on Information Theory 37, 328 (1991) DOI
[4]
R. Koetter and F. Kschischang, “Coding for Errors and Erasures in Random Network Coding”, (2008) arXiv:cs/0703061
[5]
D. Silva, F. R. Kschischang, and R. Koetter, “A Rank-Metric Approach to Error Control in Random Network Coding”, IEEE Transactions on Information Theory 54, 3951 (2008) arXiv:0711.0708 DOI
[6]
R. Koetter and F. R. Kschischang, “Coding for Errors and Erasures in Random Network Coding”, IEEE Transactions on Information Theory 54, 3579 (2008) DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: maximum_rank_distance

Cite as:
“Maximum-rank distance (MRD) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/maximum_rank_distance
BibTeX:
@incollection{eczoo_maximum_rank_distance, title={Maximum-rank distance (MRD) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/maximum_rank_distance} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/maximum_rank_distance

Cite as:

“Maximum-rank distance (MRD) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/maximum_rank_distance

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/matrices/rank-metric/maximum_rank_distance.yml.