elements of gf 2 m thus c x u03b5gf 2 m x is a codeword in c if and only if c x is divisible by g x systematic encoding requires that when a data polynomial d x is encoded as a codeword c x the coefficients of d x should appear as coefficients of c x dividing x r d x by g x the following is obtained x r u00b7 d x q x u00b7 g x p x where deg p lt deg g thus c x x r u00b7 d x p x q x g x is a multiple of g x and is hence a codeword in gf 2 n addition and subtraction are the same operation they are both bit wise xor therefore when p x was subtracted from both sides of the of the above equation it can be viewed as having been added to rather than subtracted from the left hand side yielding the above formula for c x note that since deg p u2266 r u2212 1 the sum x r u00b7 d x p x is essentially a concatenation of the coefficients of d x with the coefficients of p x therefore the data symbols in d x are the coefficients of the terms in c x of degree r and higher and the parity symbols in p x are.