# Cyclic Redundancy Check Codewords Finding the Original Message

So I have the received transmitted message in polynomial form as $$T'(X) = X^{15}+X^{13}+X^8+X^7+X^4+X^3$$ and the error polynomial as $$E(X) = X^{16}+X^{13}+X^3$$ and the generator polynomial as $$G(X) = X^4+X^3+X+1.$$

I am trying to find $T$ before it is transmitted. It is my understanding that the received transmitted message is the original codeword transmitted ($T$) plus the error polynomial ($E$), but in this case $T'$ is less than $E$, so when I subtract to find $T$, I get a negative polynomial. What am I doing wrong?

## 1 Answer

You should do arithmetic modulo 2. Modulo 2, $-1$ is the same as $+1$.

• Is this the same as XOR? – Evan Bloemer Apr 20 '16 at 16:19
• Yes, the same thing. – Yuval Filmus Apr 20 '16 at 16:19
• Can you go into more detail as to why? I understand how to XOR it, but how does that add to it? How is subtraction the same thing as addition? – Evan Bloemer Apr 20 '16 at 16:21
• I recommend you review some basic pertinent material, or look for good lecture notes on CRC. – Yuval Filmus Apr 20 '16 at 16:22