I am trying to prove a couple of statements about information sets of linear codes, but i am having trouble with these proofs or i am not sure if i understand correct what i should prove. I would appreciate any hints or comments.
Information sets we define the following way: Consider a $\left [ n,k \right ]_{q}$ linear code $C$ is a subset of $k$ integers which are the coordinates of the linear independent columns of the generator matrix $G$ of this linear code.
- Every linear code has at least one information set.
I know that the generator matrix consists of $G=\left [ I_{k}\mid A \right ]$, where $I_{k}$ is the $k\times k$ identity matrix and $A$ is some $k\times (n-k)$ matrix. Of course, the columns of the identity matrix are linear independent and their coordinates form an information set. Can we say that every generator matrix $G$ could be brought in the form above, so the set $\left \{ 1,2,...,k \right \}$ is an information set?
- Consider the binary Hamming code. How many information sets has the $(2^{m}-1,m)$ Hamming code?
My guess is $m$, since $m$ is the rank of the generator matrix $G$, but how can i prove it?
Could anybody help me with these statements? Thank you in advance!