From the similar considerations one can find out an expression which will be H C^T = 0 where H C and 0 are all vectors H : (n-k) x n parity-check matrix. H C^G therefore will be equal to 0. Typically one tries to find out what the values of x matrix is in a particular application.
Now we transmit a code vector, which is ^T, the channel introduces an error to this particular code vector and so, when we receive a word, then we can say that R equals to C plus E. Now we have already seen that H C^T equals to 0, that means when H R^T is equal to 0 we will say that no error has occurred. But if H R^T is not equal to 0 we will say that an error has occurred, or errors have occurred and this value of H R^T is called S^T or syndrome, where S is the syndrome. In general, it is easy to find syndrome using some kind of a LFSR and the associated circuitry.