Maximum-rank distance (MRD) code[1][2][3]

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) \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.

Zoo code information

Internal code ID: maximum_rank_distance

Your contribution is welcome!

on github.com (edit & pull request)

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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} }
Permanent link:
https://errorcorrectionzoo.org/c/maximum_rank_distance

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]
Ralf Koetter and Frank Kschischang, “Coding for Errors and Erasures in Random Network Coding”. 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). DOI; 0711.0708
[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

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/tree/main/codes/classical/matrices/maximum_rank_distance.yml.