Rank Codes

Synonyms

Rank-metric codes

Related Concepts

Algebraic Coding Theory; Code-Based Cryptography; McElice Public Key Cryptosystem

Definition

There exist two representations of Rank codes: matrix representation and vector representation.

In matrix representation, rank codes are defined as subsets of a normed space $\left \{{\mathbb{F}}_{q}^{N\times n},\ \textrm{ Rk}\right \}$ of N ×n matrices over a finite (base) field ${\textrm{ F}}_{q}$, where the norm of a matrix $M \in {\mathbb{F}}_{q}^{N\times n}$ is defined to be the algebraic rank $\textrm{ Rk}(M)$ of this matrix over ${\mathbb{F}}_{q}$. The rank distance between two matrices ${M}_{1} and {M}_{2}$ is the rank of their difference $\textrm{ Rk}({M}_{1} - {M}_{2})$. The rank distance of a matrix rank code $\mathcal{M}\subset {\mathbb{F}}_{q}^{N\times n}$ is defined as the minimal pairwise distance: $d(\mathcal{M}) = d =\min (\textrm{ Rk}({M}_{i} - {M}_{j}) : {M}_{i},{M}_{j} \in \mathcal{M},\ i\neq j)$. In vector...