Rank-metric code

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 \(\leq \lfloor \frac{d-1}2 \rfloor\).

Decoding

Polynomial-reconstruction Berlekamp-Welch based decoder [2].Berlekamp-Massey based decoder [3].

Notes

See Ref. [4] for a discussion of MacWilliams identities and the relationship between rank metric and Gabidulin codes.

Parent

Matrix-based code

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. [4]). The reverse is not always true since Gabidulin codes are not always \(GF(q^N)\)-linear (see Rm. 16 in Ref. [4]).
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]: P. Loidreau, “A Welch–Berlekamp Like Algorithm for Decoding Gabidulin Codes”, Coding and Cryptography 36 (2006). DOI
[3]: 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
[4]: Alberto Ravagnani, “Rank-metric codes and their duality theory”. 1410.1333

Cite as:

“Rank-metric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/rank_metric

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/matrices/rank_metric.yml.