# Rank-metric code[1]

## Description

Also called a Delsarte code. Each codeword is a matrix over \(GF(q)\), with codewords forming a \(GF(q)\)-linear subspace, and with the metric being the rank of the difference of matrices. The distance \(d\) is the minimum rank of all nonzero matrices in the code. Rank-metric codes on \(n\times m\) matrices are denoted as \([n\times m,k,d]_q\).

The number of codewords satisfies \(k \leq \max(n, m) M\), where \(M\) is the maximum rank of all matrices in the code. Codes that achieve this bound with equality are called Delsarte optimal anticodes.

## Protection

Protects against errors with rank \(\leq \lfloor \frac{d-1}2 \rfloor\).

The complexity of decoding rank-metric codes is unknown but expected to be harder than that of binary linear codes [2].

## Decoding

## Realizations

## Notes

## Parent

## Children

- Gabidulin code — Gabidulin codes over \(GF(q^N)\), when expressed as matrices over \(GF(q)\), are rank-metric codes (see Def. 14 in Ref. [8]). The reverse is not always true since Gabidulin codes are not always \(GF(q^N)\)-linear [8; Rm. 16].
- Low-rank parity-check (LRPC) code
- Maximum-rank distance (MRD) code

## 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]
- G. Philippe and Z. Gilles, “On the hardness of the decoding and the minimum distance problems for rank codes”, (2014) arXiv:1404.3482
- [3]
- P. Loidreau, “A Welch–Berlekamp Like Algorithm for Decoding Gabidulin Codes”, Coding and Cryptography 36 (2006) DOI
- [4]
- G. Richter and S. Plass, “Fast decoding of rank-codes with rank errors and column erasures”, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings. DOI
- [5]
- P. Gaborit et al., “Identity-Based Encryption from Codes with Rank Metric”, Advances in Cryptology – CRYPTO 2017 194 (2017) DOI
- [6]
- P. Lefèvre, P. Carré, and P. Gaborit, “Application of rank metric codes in digital image watermarking”, Signal Processing: Image Communication 74, 119 (2019) DOI
- [7]
- D. Silva and F. R. Kschischang, “Rank-Metric Codes for Priority Encoding Transmission with Network Coding”, 2007 10th Canadian Workshop on Information Theory (CWIT) (2007) DOI
- [8]
- A. Ravagnani, “Rank-metric codes and their duality theory”, (2015) arXiv:1410.1333

## Page edit log

- Mazin Karjikar (2023-01-16) — most recent
- Victor V. Albert (2023-01-16)
- Mazin Karjikar (2022-12-30)
- Victor V. Albert (2022-05-25)
- Thomas Wrona (2022-05-18)

## Cite as:

“Rank-metric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/rank_metric