Gabidulin code[1,2] 


Also called a vector rank-metric code. A linear code over \(GF(q^N)\) that corrects errors over rank metric instead of the traditional Hamming distance. Every element \(GF(q^N)\) can be written as an \(N\)-dimensional vector with coefficients in \(GF(q)\), and the rank of a set of elements is rank of the matrix formed by their coefficients.

Given \(X^n=\text{span}\{x_i\}\), an\(n\)-dimensional vector space over \(GF(q^N)\) (where \(q\) is a power of a prime number), the rank metric \(d(x, y)\) is defined via the rank norm \(r(x, q) = \mathrm{rank}(A(x))\), where \begin{align} A(x) = \begin{pmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ a_{N1} & a_{N2} & \ldots & a_{Nn}~, \end{pmatrix} \tag*{(1)}\end{align} and \(x_i = a_{1i} u_1 + a_{2i} u_2 + \ldots + a_{Ni}u_N \) for some fixed basis \(\{u_i\}_{i=1}^N\).


Set of vectors \(\{x_1, x_2, \ldots, x_M\}\) determines a rank code with distance \(d=\min d(x_i, x_j)\). The code with distance \(d\) corrects all errors with rank of the error not greater than \(\lfloor (d-1)/2\rfloor\).


Fast decoder based on a transform-domain approach [3].Algebraic list decoder that decodes up to the Singleton bound [4].


Public-key cryptosystems [5,6].Digital watermarking. The Gabidulin code provides efficient correction against luminance tampering and image-slicing distortion due to the consistency of the rank against alterations such as column swapping [7].


  • Rank-metric 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].


  • Maximum-rank distance (MRD) code — Gabidulin codes over \(GF(q^N)\) with maximum rank-distance, when expressed as matrices over \(GF(q)\), are MRD codes.
  • Linear \(q\)-ary code — Gabidulin codes over \(GF(q^N)\), when expressed as vectors over \(GF(q^N)\), are linear \(q\)-ary codes.


E. M. Gabidulin, Theory of Codes with Maximum Rank Distance, Problemy Peredachi Informacii, Volume 21, Issue 1, 3–16 (1985)
R. M. Roth, “Maximum-rank array codes and their application to crisscross error correction”, IEEE Transactions on Information Theory 37, 328 (1991) DOI
D. Silva and F. R. Kschischang, “Fast encoding and decoding of Gabidulin codes”, 2009 IEEE International Symposium on Information Theory (2009) arXiv:0901.2483 DOI
V. Guruswami and C. Xing, “List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound”, Proceedings of the forty-fifth annual ACM symposium on Theory of Computing (2013) DOI
T. P. Berger and P. Loidreau, “How to Mask the Structure of Codes for a Cryptographic Use”, Designs, Codes and Cryptography 35, 63 (2005) DOI
T. Lau and C. Tan, “A New Technique in Rank Metric Code-Based Encryption”, Cryptography 2, 32 (2018) DOI
P. Lefevre, P. Carre, and P. Gaborit, “Watermarking and Rank Metric Codes”, 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2018) DOI
A. Ravagnani, “Rank-metric codes and their duality theory”, (2015) arXiv:1410.1333
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Zoo Code ID: gabidulin

Cite as:
“Gabidulin code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_gabidulin, title={Gabidulin code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Gabidulin code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.