{0010111, 1110010, 1100101, 1010001, 0100011, 0101110, 0001101, 1111111, 1011100, 1101000, 0111001, 0000000, 1000110, 1001011, 0110100, 0011010}
i was given 16 codewords and need to find generator matrix. i know generator matrix contains code word that linear independent, but how can i know is it linear independent or not? and also from 16 code words, i know k=4 which is the row in generator matrix,
if i add 0010111,+1110010=1100101 , so i cant use 3 so 1,2,4,6 can be generator matrix, am i right? i just need to add and as long as there is no same codeword that produces by adding two codeword together in matrices, then it linear independent? and also there can be many possibilities too?
If you are given only the set of codewords (16 here), you only need to find 4 (because $2^4=16$) linearly independent codewords. Yes, there are many possibilities for doing that, you have a lot of freedom. A simple way here is to look for 4 codewords that begin (or end) with the 4 possible $2-$tuples (00 01 10 00) (you should understand that that automatically implies they're LI).
If you instead are given not only the set of codewords but the full code, i.e. the mapping from raw vectors to codewords, then you have no such freedom, and there is single generator matrix.