Maximum-rank distance (MRD) code[13] 

Also known as Optimal rank-distance code.

Description

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} becomes an equality.

Realizations

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

Parent

Cousins

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
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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} }
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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.