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In coding theory, rank 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 = ⌊ / 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 {\displaystyle GF} similar to Reed–Solomon code.
The rank of the vector over G F {\displaystyle GF} is the maximum number of linearly independent components over G F {\displaystyle GF}. The rank distance between two vectors over G F {\displaystyle GF} 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.