Rank error-correcting code
In coding theory, rank codes (also called Gabidulin codes) are non-binary over not but rank metric. They described a systematic way of building codes that could detect and correct multiple 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 , it can correct up to d − 1 known erasures.
A rank code is an algebraic linear code over the finite field <math>GF(q^N)</math> similar to Reed–Solomon code.
The rank of the vector over <math>GF(q^N)</math> is the maximum number of linearly independent components over <math>GF(q)</math>. The rank distance between two vectors over <math>GF(q^N)</math> 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.
Contents
Rank metric
Let <math>X^n</math> be an n-dimensional vector space over the <math>GFleft( {q^N } right)</math>, where <math>q</math> is a power of a prime, <math>N</math> is an integer and <math>left(u_1, u_2, dots, u_Nright)</math> with <math>u_i in GF(q)</math> is a base of the vector space over the field <math>GFleft( {q} right)</math>.
Every element <math>x_i in GFleft( {q^N } right)</math> can be represented as <math>x_i = a_{1i}u_1 + a_{2i}u_2 + dots + a_{Ni}u_N</math>. Hence, every vector <math>vec x = left( {x_1, x_2, dots, x_n } right)</math> over <math>GFleft( {q^N } right)</math> can be written as matrix:
- <math>
vec x = left| {begin{array}{*{20}c}
a_{1,1} & a_{1,2} & ldots & a_{1,n} \ a_{2,1} & a_{2,2} & ldots & a_{2,n} \ ldots & ldots & ldots & ldots \ a_{N,1} & a_{N,2} & ldots & a_{N,n} end{array}} right| </math>
Rank of the vector <math>vec x</math> over the field <math>GFleft( {q^N} right)</math> is a rank of the corresponding matrix <math>Aleft( {vec x} right)</math> over the field <math>GFleft( {q} right)</math> denoted by <math>rleft( {vec x; q} right)</math>.
The set of all vectors <math>vec x</math> is a space <math>X^n = A_N^n</math>. The map <math>vec x to rleft( vec x; q right)</math>) defines a norm over <math>X^n</math> and a rank metric:
- <math>
dleft( {vec x;vec y} right) = rleft( {vec x - vec y;q} right) </math>
Rank code
A set <math>{x_1, x_2, dots, x_n}</math> of vectors from <math>X^n</math> is called a code with code distance <math>d = min dleft( x_i ,x_j right)</math> and a k-dimensional subspace of <math>X^n</math> – a linear (n, k)-code with distance <math>d leq n – k + 1</math>.
Generating matrix
There is known the only construction of rank code, which is a maximum rank distance MRD-code with d = n − k + 1.
Let’s define a Frobenius power <math>[i]</math> of the element <math>x in GF(q^N)</math> as
- <math>
x^{[i]} = x^{q^{i mod N}}. , </math>
Then, every vector <math>vec g = (g_1, g_2, dots, g_n), ~ g_i in GF(q^N), ~ n leq N</math>, linearly independent over <math>GF(q)</math>, defines a generating matrix of the MRD (n, k, d = n − k + 1)-code.
- <math>
G = left| {begin{array}{*{20}c}
g_1 & g_2 & dots & g_n \ g_1^{[m]} & g_2^{[m]} & dots & g_n^{[m]} \ g_1^{[2 m]} & g_2^{[2 m]} & dots & g_n^{[2 m]} \ dots & dots & dots & dots \ g_1^{[k m]} & g_2^{[k m]} & dots & g_n^{[k m]} end{array}} right|, </math>
where <math>gcd(m,N) = 1</math>.
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 .
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