Abstract Algebra: Theory and Applications

BCH codes have very attractive error correction algorithms. Let $C$ be a BCH code in $R_n\text{,}$ and suppose that a code polynomial $c(t)=c_0+c_{1}t+\cdots +c_{n-1}{t}^{n-1}$ is transmitted. Let $w(t)=w_0+w_{1}t+\cdots w_{n-1}{t}^{n-1}$ be the polynomial in $R_n$ that is received. If errors have occurred in bits $a_1,\dots ,a_k\text{,}$ then $w(t)=c(t)+e(t)\text{,}$ where $e(t)=t^{a_1}+t^{a_2}+\cdots +t^{a_k}$ is the error polynomial. The decoder must determine the integers $a_i$ and then recover $c(t)$ from $w(t)$ by flipping the $a_i$th bit. From $w(t)$ we can compute $w({\omega }^i)=s_i$ for $i=1,\dots ,2r\text{,}$ where $\omega$ is a primitive $n$th root of unity over ${\mathbb{Z}}_2\text{.}$ We say the syndrome of $w(t)$ is $s_1,\dots ,s_{2r}\text{.}$