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} \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].


  • Linear \(q\)-ary code
  • 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. [4]). The reverse is not always true since Gabidulin codes are not always \(GF(q^N)\)-linear (see Rm. 16 in Ref. [4]).


Zoo code information

Internal code ID: gabidulin

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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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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). DOI; 0901.2483
Alberto Ravagnani, “Rank-metric codes and their duality theory”. 1410.1333

Cite as:

“Gabidulin code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.