Rank error-correcting code
In coding theory, rank codes (also called Gabidulin codes) are non-binary linear error-correcting codes over not Hamming but rank metric. They described a systematic way of building codes that could detect and correct multiple random rank errors. By adding redundancy with coding k-symbol word to a n-symbol word, a rank code can correct any errors of rank up to t = ⌊ (d − 1) / 2 ⌋, where d is a code distance. As an erasure code, it can correct up to d − 1 known erasures.

A rank code is an algebraic linear code over the finite field  similar to Reed–Solomon code.

The rank of the vector over  is the maximum number of linearly independent components over . The rank distance between two vectors over  is the rank of the difference of these vectors.

The rank code corrects all errors with rank of the error vector not greater than t.

Rank metric 
Let  be an n-dimensional vector space over the finite field , where  is a power of a prime and  is a positive integer. Let  , with , be a base of  as a vector space over the field .

Every element  can be represented as . Hence, every vector  over  can be written as matrix:

 

Rank of the vector  over the field  is a rank of the corresponding matrix   over the field  denoted by .

The set of all vectors  is a space . The map ) defines a norm over  and a rank metric:

Rank code
A set  of vectors from  is called a code with code distance  .  If the set also forms a k-dimensional subspace of , then it is called a linear (n, k)-code with distance .  Such a linear rank metric code always satisfies the Singleton bound  with equality.

Generating matrix 
There are several known constructions of rank codes, which are maximum rank distance (or MRD) codes with d = n − k + 1.
The easiest one to construct is known as the (generalized) Gabidulin code, it was discovered first by Delsarte (who called it a  Singleton system) and later by Gabidulin  (and Kshevetskiy  ).

Let's define a Frobenius power  of the element  as

 

Then, every vector , linearly independent over , defines a generating matrix of the MRD (n, k, d = n − k + 1)-code.

 

where .

Applications 
There are several proposals for public-key cryptosystems based on rank codes. However, most of them have been proven insecure (see e.g. Journal of Cryptology,
April 2008).

Rank codes are also useful for error and erasure correction in network coding.

See also
 Linear code
 Reed–Solomon error correction
 Berlekamp–Massey algorithm
 Network coding

Notes

References

External links
 MATLAB implementation of a Rank–metric codec

Error detection and correction
Coding theory