Rank error-correcting code


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 = ⌊ / 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.

Rank metric

Let be an n-dimensional vector space over the finite field, where is a power of a prime and is a positive integer. Let , with, be a base of as a 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:

Rank code

A set of vectors from is called a code with code distance . If the set also forms a k-dimensional subspace of, then it is called a linear -code with distance. Such a linear rank metric code always satisfies the Singleton bound.

Generating matrix

There are several known constructions of rank codes, which are maximum rank distance codes with d = n − k + 1.
The easiest one to construct is known as the Gabidulin code, it was discovered first by Delsarte and later by Gabidulin.
Let's define a Frobenius power of the element as
Then, every vector, linearly independent over, defines a generating matrix of the MRD -code.
where.

Applications

There are several proposals for public-key cryptosystems based on rank codes. However, most of them have been proven insecure.
Rank codes are also useful for error and erasure correction in network coding.