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| name          = Rank codes
| image          =
| image_caption  =
| hierarchy      = [[Linear block code]]<br>[[Rank error correcting code|Rank code]]
| block_length  = ''n''
| message_length = ''k''
| distance      = ''n'' &minus; ''k'' + 1
| alphabet_size  = ''Q'' = ''q''<sup>''N''</sup>&nbsp; (''q'' prime)
| notation      = [''n'', ''k'', ''d'']-code
| decoding      = [[Berlekamp–Massey algorithm|Berlekamp–Massey]]<br>[[Euclidean algorithm|Euclidean]]<br>''with Frobenius poynomials''
}}
 
In [[coding theory]], '''rank codes''' are non-binary<ref name="non-binary">Codes for which each input symbol is from a set of size greater than 2.</ref> [[linear code|linear]] [[error-correcting code]]s over not [[Hamming metric|Hamming]] but ''rank'' metric. They described a systematic way of building codes that could detect and correct multiple [[random error|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''&nbsp;=&nbsp;⌊&nbsp;(''d''&nbsp;&minus;&nbsp;1)&nbsp;/&nbsp;2&nbsp;⌋, where ''d'' is a code distance. As an [[erasure code]], it can correct up to ''d'' &minus; 1 known erasures.
 
A '''rank code''' is an algebraic linear code over the finite field <math>GF(q^N)</math> similar to [[Reed–Solomon error correction|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&nbsp;''t''.
 
== Rank metric ==
Let <math>X^n</math> — ''n''-dimensional vector space over the [[finite field]] <math>GF\left( {q^N } \right)</math>, where <math>q</math> is a prime number, <math>N</math> is a power and <math>\left(u_1, u_2, \dots, u_N\right)</math> with <math>u_i \in GF(q)</math> is a base of the vector space over the field <math>GF\left( {q} \right)</math>.
 
Every element <math>x_i  \in GF\left( {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>GF\left( {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>GF\left( {q^N} \right)</math> is a rank of the corresponding matrix  <math>A\left( {\vec x} \right)</math> over the field <math>GF\left( {q} \right)</math> denoted by <math>r\left( {\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 r\left( \vec x; q \right)</math>) defines a norm over <math>X^n</math> and a ''rank metric'':
 
: <math>
    d\left( {\vec x;\vec y} \right) = r\left( {\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 d\left( 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''&nbsp;=&nbsp;''n''&nbsp;&minus;&nbsp;''k''&nbsp;+&nbsp;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''&nbsp;=&nbsp;''n''&nbsp;&minus;&nbsp;''k''&nbsp;+&nbsp;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<ref>http://link.springer.com/article/10.1007%2Fs00145-007-9003-9)</ref>).
 
Rank codes are also suitable for [[network coding]].
 
==See also==
* [[Linear code]]
* [[Reed–Solomon error correction]]
* [[Berlekamp–Massey algorithm]]
* [[Network coding]]
 
==Notes==
{{Reflist}}
 
==References==
*{{Citation
|first=Ernst M.
|last=Gabidulin
|title=Theory of codes with maximum rank distance
|journal=Problems of Information Transmission
|volume=21
|issue=1
|year=1985
|pages=1–12
|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=967&option_lang=eng
}}
*{{Citation
|first= Alexander
|last= Kshevetskiy
|first2= Ernst M.
|last2= Gabidulin
|title=The new construction of rank codes
|journal=Proceedings of IEEE International Symposium on Information Theory (ISIT) 2005
|year=4–9 Sept. 2005
|pages=2105–2108
|isbn = 0-7803-9151-9
|url=http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1523717
}}
*{{Citation
|first= Ernst M.
|last= Gabidulin
|first2= Nina I.
|last2= Pilipchuk
|title=A new method of erasure correction by rank codes
|journal=Proceedings of the 2003 IEEE International Symposium on Information Theory
|date=June 29–July 4, 2003
|pages=423
|isbn = 0-7803-7728-1
|url=http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1228440
}}
 
[[Category:Error detection and correction]]
[[Category:Coding theory]]

Revision as of 09:04, 12 February 2014

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