Rank error-correcting code

In <a href="/facts/Coding_theory/SBqStoGL">coding theory</a>, rank codes (also called Gabidulin codes) are non-binary <a href="/facts/Linear_code/HoFI3eyF">linear</a> <a href="/facts/Error-correcting_code/cLUTUJrA">error-correcting codes</a> over not <a href="/facts/Hamming_metric/MgdtbYPo">Hamming</a> but rank metric. They described a systematic way of building codes that could detect and correct multiple <a href="/facts/Random_error/q7pa7bRt">random</a> 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 <a href="/facts/Erasure_code/ea1C1z2X">erasure code</a>, 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 <a href="/facts/Reed%25E2%2580%2593Solomon_error_correction/HPc3Qj6N">Reed–Solomon code</a>.
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 error-correcting code open-in-new

Rank error-correcting code