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 G F ( q N ) {\displaystyle GF(q^{N})} similar to Reed–Solomon code. The rank of the vector over G F ( q N ) {\displaystyle GF(q^{N})} is the maximum number of linearly independent components over G F ( q ) {\displaystyle GF(q)} . The rank distance between two vectors over G F ( q N ) {\displaystyle GF(q^{N})} 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 X n {\displaystyle X^{n}} be an n-dimensional vector space over the finite field G F ( q N ) {\displaystyle GF\left({q^{N}}\right)} , where q {\displaystyle q} is a power of a prime and N {\displaystyle N} is a positive integer. Let ( u 1 , u 2 , … , u N ) {\displaystyle \left(u_{1},u_{2},\dots ,u_{N}\right)} , with u i ∈ G F ( q N ) {\displaystyle u_{i}\in GF(q^{N})} , be a base of G F ( q N ) {\displaystyle GF\left({q^{N}}\right)} as a vector space over the field G F ( q ) {\displaystyle GF\left({q}\right)} . Every element x i ∈ G F ( q N ) {\displaystyle x_{i}\in GF\left({q^{N}}\right)} can be represented as x i = a 1 i u 1 + a 2 i u 2 + ⋯ + a N i u N {\displaystyle x_{i}=a_{1i}u_{1}+a_{2i}u_{2}+\dots +a_{Ni}u_{N}} . Hence, every vector x → = ( x 1 , x 2 , … , x n ) {\displaystyle {\vec {x}}=\left({x_{1},x_{2},\dots ,x_{n}}\right)} over G F ( q N ) {\displaystyle GF\left({q^{N}}\right)} can be written as matrix:
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