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} become an equality.
Realizations
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
- Victor V. Albert (2022-05-25) — most recent
- Victor V. Albert (2021-12-16)
- Marianna Podzorova (2021-12-13)
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