Rank error-correcting code
|Hierarchy||Linear block code
|Distance||n − k + 1|
|Alphabet size||Q = qN (q prime)|
|Notation||[n, k, d]-code|
with Frobenius poynomials
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 = ⌊ (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 similar to Reed–Solomon code.
The rank of the vector over is the maximum number of linearly independent components over . The rank distance between two vectors over 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.
Let — n-dimensional vector space over the finite field , where is a prime number, is a power and with is a base of the vector space over the field .
Every element can be represented as . Hence, every vector over can be written as matrix:
Rank of the vector over the field is a rank of the corresponding matrix over the field denoted by .
The set of all vectors is a space . The map ) defines a norm over and a rank metric:
A set of vectors from is called a code with code distance and a k-dimensional subspace of – a linear (n, k)-code with distance .
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 of the element as
Then, every vector , linearly independent over , defines a generating matrix of the MRD (n, k, d = n − k + 1)-code.
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 suitable for network coding.
- Codes for which each input symbol is from a set of size greater than 2.
- Gabidulin, Ernst M. (1985), "Theory of codes with maximum rank distance", Problems of Information Transmission 21 (1): 1–12
- Kshevetskiy, Alexander; Gabidulin, Ernst M. (4–9 Sept. 2005), "The new construction of rank codes", Proceedings of IEEE International Symposium on Information Theory (ISIT) 2005: 2105–2108, ISBN 0-7803-9151-9
- Gabidulin, Ernst M.; Pilipchuk, Nina I. (June 29–July 4, 2003), "A new method of erasure correction by rank codes", Proceedings of the 2003 IEEE International Symposium on Information Theory: 423, ISBN 0-7803-7728-1