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In coding theory, rank codes (also called Gabidulin 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 ${\displaystyle GF(q^{N})}$ similar to Reed–Solomon code.

The rank of the vector over ${\displaystyle GF(q^{N})}$ is the maximum number of linearly independent components over ${\displaystyle GF(q)}$. The rank distance between two vectors over ${\displaystyle GF(q^{N})}$ 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 metric

Let ${\displaystyle X^{n}}$ be an n-dimensional vector space over the finite field ${\displaystyle GF\left({q^{N}}\right)}$, where ${\displaystyle q}$ is a power of a prime and ${\displaystyle N}$ is a positive integer. Let ${\displaystyle \left(u_{1},u_{2},\dots ,u_{N}\right)}$, with ${\displaystyle u_{i}\in GF(q^{N})}$, be a base of ${\displaystyle GF\left({q^{N}}\right)}$ as a vector space over the field ${\displaystyle GF\left({q}\right)}$.

Every element ${\displaystyle x_{i}\in GF\left({q^{N}}\right)}$ can be represented as ${\displaystyle x_{i}=a_{1i}u_{1}+a_{2i}u_{2}+\dots +a_{Ni}u_{N}}$. Hence, every vector ${\displaystyle {\vec {x}}=\left({x_{1},x_{2},\dots ,x_{n}}\right)}$ over ${\displaystyle GF\left({q^{N}}\right)}$ can be written as matrix:

${\displaystyle {\vec {x}}=\left\|{\begin{array}{*{20}c}a_{1,1}&a_{1,2}&\ldots &a_{1,n}\\a_{2,1}&a_{2,2}&\ldots &a_{2,n}\\\ldots &\ldots &\ldots &\ldots \\a_{N,1}&a_{N,2}&\ldots &a_{N,n}\end{array}}\right\|}$

Rank of the vector ${\displaystyle {\vec {x}}}$ over the field ${\displaystyle GF\left({q^{N}}\right)}$ is a rank of the corresponding matrix ${\displaystyle A\left({\vec {x}}\right)}$ over the field ${\displaystyle GF\left({q}\right)}$ denoted by ${\displaystyle r\left({{\vec {x}};q}\right)}$.

The set of all vectors ${\displaystyle {\vec {x}}}$ is a space ${\displaystyle X^{n}=A_{N}^{n}}$. The map ${\displaystyle {\vec {x}}\to r\left({\vec {x}};q\right)}$) defines a norm over ${\displaystyle X^{n}}$ and a rank metric:

${\displaystyle d\left({{\vec {x}};{\vec {y}}}\right)=r\left({{\vec {x}}-{\vec {y}};q}\right)}$

Rank code

A set ${\displaystyle \{x_{1},x_{2},\dots ,x_{n}\}}$ of vectors from ${\displaystyle X^{n}}$ is called a code with code distance ${\displaystyle d=\min d\left(x_{i},x_{j}\right)}$. If the set also forms a k-dimensional subspace of ${\displaystyle X^{n}}$, then it is called a linear (n, k)-code with distance ${\displaystyle d}$. Such a linear rank metric code always satisfies the Singleton bound ${\displaystyle d\leq n-k+1}$ with equality.

Generating matrix

There are several known constructions of rank codes, which are maximum rank distance (or MRD) codes with d = n − k + 1. The easiest one to construct is known as the (generalized) Gabidulin code, it was discovered first by Delsarte (who called it a Singleton system) and later by Gabidulin (and Kshevetskiy ).

Let's define a Frobenius power ${\displaystyle }$ of the element ${\displaystyle x\in GF(q^{N})}$ as

${\displaystyle x^{}=x^{q^{i\mod N}}.\,}$

Then, every vector ${\displaystyle {\vec {g}}=(g_{1},g_{2},\dots ,g_{n}),~g_{i}\in GF(q^{N}),~n\leq N}$, linearly independent over ${\displaystyle GF(q)}$, defines a generating matrix of the MRD (n, k, d = n − k + 1)-code.

${\displaystyle G=\left\|{\begin{array}{*{20}c}g_{1}&g_{2}&\dots &g_{n}\\g_{1}^{}&g_{2}^{}&\dots &g_{n}^{}\\g_{1}^{}&g_{2}^{}&\dots &g_{n}^{}\\\dots &\dots &\dots &\dots \\g_{1}^{}&g_{2}^{}&\dots &g_{n}^{}\end{array}}\right\|,}$

where ${\displaystyle \gcd(m,N)=1}$.

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 network coding.

See also

• Linear code
• Reed–Solomon error correction
• Berlekamp–Massey algorithm
• Network coding

Notes

1. Codes for which each input symbol is from a set of size greater than 2.
2. Gabidulin, Ernst M. (1985). "Theory of codes with maximum rank distance". Problems of Information Transmission. 21 (1): 1–12.
3. Kshevetskiy, Alexander; Gabidulin, Ernst M. (4–9 September 2005). "The new construction of rank codes". Proceedings. International Symposium on Information Theory, 2005. ISIT 2005. pp. 2105–2108. doi:10.1109/ISIT.2005.1523717. ISBN 978-0-7803-9151-2. S2CID 11679865.
4. Overbeck, R. (2008). "Structural Attacks for Public Key Cryptosystems based on Gabidulin Codes". Journal of Cryptology. 21 (2): 280–301. doi:10.1007/s00145-007-9003-9. S2CID 2393853.

References

• 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 September 2005). "The new construction of rank codes". Proceedings. International Symposium on Information Theory, 2005. ISIT 2005. pp. 2105–2108. doi:10.1109/ISIT.2005.1523717. ISBN 978-0-7803-9151-2. S2CID 11679865.
• Gabidulin, Ernst M.; Pilipchuk, Nina I. (June 29 – July 4, 2003). "A new method of erasure correction by rank codes". IEEE International Symposium on Information Theory, 2003. Proceedings. p. 423. doi:10.1109/ISIT.2003.1228440. ISBN 978-0-7803-7728-8. S2CID 122552232.

External links

• MATLAB implementation of a Rank–metric codec

• Error detection and correction
• Coding theory