Rank error-correcting code

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:

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

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.

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 [2] (and Kshevetskiy [3] ).

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

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

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.

${\displaystyle G=\left\|{\begin{array}{*{20}c}g_{1}&g_{2}&\dots &g_{n}\\g_{1}^{[m]}&g_{2}^{[m]}&\dots &g_{n}^{[m]}\\g_{1}^{[2m]}&g_{2}^{[2m]}&\dots &g_{n}^{[2m]}\\\dots &\dots &\dots &\dots \\g_{1}^{[(k-1)m]}&g_{2}^{[(k-1)m]}&\dots &g_{n}^{[(k-1)m]}\end{array}}\right\|,}$

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

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[4]).

Rank codes are also useful for error and erasure correction in network coding.

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

• 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.

• MATLAB implementation of a Rank–metric codec