Editing BCH code (section)

=== Systematic encoding: The message as a prefix ===
A systematic code is one in which the message appears verbatim somewhere within the codeword. Therefore, systematic BCH encoding involves first embedding the message polynomial within the codeword polynomial, and then adjusting the coefficients of the remaining (non-message) terms to ensure that <math>s(x)</math> is divisible by <math>g(x)</math>.

This encoding method leverages the fact that subtracting the remainder from a dividend results in a multiple of the divisor. Hence, if we take our message polynomial <math>p(x)</math> as before and multiply it by <math>x^{n-k}</math> (to "shift" the message out of the way of the remainder), we can then use [[Euclidean division]] of polynomials to yield:
:<math>p(x)x^{n-k} = q(x)g(x) + r(x)</math>

Here, we see that <math>q(x)g(x)</math> is a valid codeword. As <math>r(x)</math> is always of degree less than <math>n-k</math> (which is the degree of <math>g(x)</math>), we can safely subtract it from <math>p(x)x^{n-k}</math> without altering any of the message coefficients, hence we have our <math>s(x)</math> as
:<math>s(x) = q(x)g(x) = p(x)x^{n-k} - r(x)</math>

Over <math>GF(2)</math> (i.e. with binary BCH codes), this process is indistinguishable from appending a [[cyclic redundancy check]], and if a systematic binary BCH code is used only for error-detection purposes, we see that BCH codes are just a generalization of the [[mathematics of cyclic redundancy checks]].

The advantage to the systematic coding is that the receiver can recover the original message by discarding everything after the first <math>k</math> coefficients, after performing error correction.