# Coding theory, rank metric codes

Let $S$ be a subset in $Mat_{k\times n}(\mathbb{F}_q)$ ($k\times n$ matrices with entries in $\mathbb{F}_q$) and $d(A,B)= rank(A-B)$, $dist(S)= \min\{d(A,B) \mid A,B \in S, A\neq B\}$

What is maximal size of set $S$ such that $dist(S)=d$ and $\mathbb{F}_q,k,n$ are fixed?

I tried for $k=2,n=2, \mathbb{F}_q=\mathbb{F}_2=\{0,1\}$ when $dist(S)=1$, I got maximal size of $S$ is $2$.

But is there any easy way or logic doing this for any $k$ and $n$ over $\mathbb{F}_2$?

Can anyone explain this in details please?

## 1 Answer

Actually, Rank Metric Codes is currently an active research area in coding theory. If you are truly interested in this topic, I think you can start with Dr Gadouleau's series works and the references therein. Many of his papers are available on arXiv.

• Thank you for the link. Also can you please suggest me, any articles or may be book where I can read in details about rank metric codes? – user134906 Mar 12 '14 at 10:48
• @user134906 Items 4, 5, 6 in the second link can be a start. – Fei Gao Mar 12 '14 at 12:55