Exercises 17.5 Additional Exercises: Error Correction for BCH Codes
1.
Show that \(w(t)\) is a code polynomial if and only if \(s_i = 0\) for all \(i\text{.}\)
2.
Show that
\begin{equation*}
s_i = w( \omega^i) = e( \omega^i) = \omega^{i a_1} + \omega^{i a_2} + \cdots + \omega^{i a_k}
\end{equation*}
for \(i = 1, \ldots, 2r\text{.}\) The error-locator polynomial is defined to be
\begin{equation*}
s(x) = (x + \omega^{a_1})(x + \omega^{a_2}) \cdots (x + \omega^{a_k})\text{.}
\end{equation*}
3.
Recall the \((15,7)\)-block BCH code in Example 17.19. By Theorem 16.13, this code is capable of correcting two errors. Suppose that these errors occur in bits \(a_1\) and \(a_2\text{.}\) The error-locator polynomial is \(s(x) = (x + \omega^{a_1})(x + \omega^{a_2})\text{.}\) Show that
\begin{equation*}
s(x) = x^2 + s_1 x + \left( s_1^2 + \frac{s_3}{s_1} \right)\text{.}
\end{equation*}
4.
Let \(w(t) = 1 + t^2 +t^4 + t^5 + t^7 + t^{12} + t^{13}\text{.}\) Determine what the originally transmitted code polynomial was.