1.
Consider a binary, narrow-sense, primitive BCH code \(C\) of length \(15\) with designed distance \(5\) using a primitive element \(\alpha\) which is a root of \(x^4+x+1\text{.}\)
(a)
Compute \(g(x)\text{.}\)
Hint.
Solution.
As a narrow-sense code we take \(b=1\) as our starting exponent and since the designed distance is \(5\) we need four consecutive roots \(\alpha,\alpha^2,\alpha^3,\alpha^4\text{.}\) Now the conjugacy classes of roots are
\begin{align*}
\amp \{\alpha,\alpha^2,\alpha^4,\alpha^8\} \\
\amp \{\alpha^3,\alpha^6,\alpha^{12},\alpha^9\}\text{.}
\end{align*}
So \(g(x)=m_{\alpha}(x)m_{\alpha^3}(x)\text{.}\)
We have \(m_{\alpha}(x)=x^4+x+1\) by our choice of \(\alpha\text{.}\) \(\alpha^3\) has order \(5\) so is a root of a non-primitive polynomial of degree \(4\text{,}\) which must be \(x^4+x^3+x^2+x+1\text{.}\) So finally
\begin{align*}
g(x) \amp =(x^4+x+1)(x^4+x^3+x^2+x+1) \\
\amp = x^8+x^7+x^6+x^4+1\text{.}
\end{align*}
(b)
(c)
Show that in this case the BCH bound holds with equality.
