Editing BCH code (section)

=== Non-systematic encoding: The message as a factor ===
The most straightforward way to find a polynomial that is a multiple of the generator is to compute the product of some arbitrary polynomial and the generator. In this case, the arbitrary polynomial can be chosen using the symbols of the message as coefficients.

:<math>s(x) = p(x)g(x)</math>

As an example, consider the generator polynomial <math>g(x)=x^{10}+x^9+x^8+x^6+x^5+x^3+1</math>, chosen for use in the (31, 21) binary BCH code used by [[POCSAG]] and others. To encode the 21-bit message {101101110111101111101}, we first represent it as a polynomial over <math>GF(2)</math>:
:<math>p(x) = x^{20}+x^{18}+x^{17}+x^{15}+x^{14}+x^{13}+x^{11}+x^{10}+x^9+x^8+x^6+x^5+x^4+x^3+x^2+1</math>
Then, compute (also over <math>GF(2)</math>):
:<math>\begin{align}
  s(x) &= p(x)g(x)\\
       &= \left(x^{20}+x^{18}+x^{17}+x^{15}+x^{14}+x^{13}+x^{11}+x^{10}+x^9+x^8+x^6+x^5+x^4+x^3+x^2+1\right)\left(x^{10}+x^9+x^8+x^6+x^5+x^3+1\right)\\
       &= x^{30}+x^{29}+x^{26}+x^{25}+x^{24}+x^{22}+x^{19}+x^{17}+x^{16}+x^{15}+x^{14}+x^{12}+x^{10}+x^9+x^8+x^6+x^5+x^4+x^2+1
\end{align}</math>
Thus, the transmitted codeword is {1100111010010111101011101110101}.

The receiver can use these bits as coefficients in <math>s(x)</math> and, after error-correction to ensure a valid codeword, can recompute <math>p(x) = s(x)/g(x)</math>