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\begin{center} {\large \textbf{Math 437 -- Homework 5}} \smallskip \\ 
Due 10:15am on Thursday, February 16, 2017  \\ \ \\
 \end{center}

Please indicate any sources you used for a given problem on the solution to that problem. 
For example, if you worked with another student to get the solution to a problem, please indicate who.
You are welcome to work together in small groups, but please try the problems on your own first and write up 
your own solutions. \\ \medskip 




\begin{prob}
Let $p\in \Z_2[x]$ be a primitive polynomial of degree $n$ and take $F=\Z_2[a]/(p(a))$.  
Suppose $e(x) \in \Z_2[x]$ has syndromes 
$s_1=e(a), s_2=e(a^2), s_3=e(a^3)$ and let 
\[
S  \ \ =  \ \ 
\begin{pmatrix}
s_1 & s_2   \\
s_2 & s_3  \\
\end{pmatrix}.
\]
Show the following: 
\begin{itemize}
\item[(a)] If $e(x)$ has one term (say $e(x) = x^d$), then the matrix $S$ has rank one.
\item[(b)] If $e(x)$ has two terms (say $e(x) = x^{d_1} + x^{d_2}$), then $S$ has rank two.  
\end{itemize}
\end{prob} \ \\


\begin{prob}
Let $C$ be the BCH code generated by the first 4 powers of $a$ in $\Z_2[a]/(a^4+a+1)$ (see page 97).  Correct each of polynomials $r(x)$ to a polynomial in $C$: 
\begin{itemize}
\item[(a)] $r(x) = 1+ x^7+x^{11} + x^{14}$ 
\item[(b)] $r(x) = x^6 + x^7+x^8 + x^{11}$
\end{itemize}
\end{prob}\ \\


\begin{prob}
Ch. 5 \#6 on page 169\\ \emph{(The table from Homework 4 \#3 may be helpful.)}
\end{prob}\ \\






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