I'm trying to understand the example given in the wikipedia explanation of the algorithm of Reed Solomon: http://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction. We have
$p(x) = 3x^6 + 2x^5 + x^4$
$g(x) = x^4 + 809x^3 + 723x^2 + 568x + 522$
$s(x) = p(x) \mod g(x) = 547x^3 + 738x^2 + 442x + 522.$
I don't understand how do we calculate $s(x)$. I thought that we just have to divide $p$ by $g$, but I enormous numbers for the remainders.
Can someone help me?
If you write $p(x) = q(x)g(x)+s(x)$ the 'normal' algorithm (i.e. working in $\mathbb{Z}[x]$) gives $$q(x)=3 x^2-2425x+1959657$$ $$s(x)=-1583610942x^3-1415456177x^2-1111819326x-1022940954.$$ If you reduce the coefficients modulo $929$ because the example is in $GF(929)$ you get $$q(x)=3x^2+362x+396$$ $$s(x)=547x^3+738x^2+442x+455.$$ Note the small typo (522 vs 455) in your $s(x)$ compared to wiki.