is a generator matrix of C iff is a control matrix of C
I am trying to prove that, ''given a C [n, k, d]-linear code, is a generator matrix of C iff is a control matrix of C''.
Firstly, I have supposed that is a generator matrix of C; we name the columns of A as ; if , I can codify this into a word of the code by taking the generator matrix and multiplying it this way: , where is the dot product of two vectors. I also know the following result:
H is a control matrix of C if the following holds: .
So, If I show , I would have show that ; this is very easy to show taking on account how H is constructed.
For the other direction, I would have to show that has, as a system of equations, as solutions all the words of the code... Here I get stucked.
Secondly, I would have to prove that, assuming is a control matrix of C, is a generator matrix of C. For this I suppose I would have to solve the system (which, by hipothesis I know has as solutions the words of the code), and then show that each of that words can be generated by G, i.e., that, given a code word, I can find a vector in such that the product of that vector by G is the code word firstly given. Nevertheless I am not sure this is the best approach...
Any help, guidance, or anything will be very helpful.