The problem goes like this.

A linear binary code is given by the control matrix H =

1 1 0 1 0 1

0 1 1 0 0 1

1 1 0 0 1 0

a) What codeword can, with at most one error, become 010110?

b) Find a word that cannot arise with at most one error in a codeword.

First, I don't understand how to construct a checkmatrix. I know it's something about binary variables, but how to get the distance and dimension right is unclear to me. Then, when you multiply the matrix with the word, how do you see where the error is? The answer is just a vector of length 3 but still you can somehow see where the error is.

Grateful is someone can clarify this to me!