Section Week 8
This is an outline of the topics we covered in the eighth week of class.
Subsection Tuesday 3/3
Coding Theorist of the Day.
Irving Reed, 1923-2012
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Work in many areas of coding, radar, signal and image processing
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Developed Reed-Solomon codes in 1960 jointly with Solomon
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Also known for Reed-Muller codes (discovered by Muller in 1954 but no decoding algorithm; efficient decoding alg. by Reed in the same year.)
Reminders.
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Exam 2 next Tuesday: modules 3 (bounds) and 4 (finite fields)
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Problem Set M4 due T, feedback before exam if turned in by class on Thursday
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Video Project Topic & Team Contract due Thursday
Quadratics in Characteristic 2.
To decode the code from last time we have to solve
\begin{equation*}
x^2+s_1x+\left(\frac{s_2}{s_1}+s_1^2\right)=0
\end{equation*}
over \(\F_{2^4}\text{.}\) How?
1. Don’t write \(\sqrt{b}\) - give an integer power of \(b\) which is this element instead. Use the fact that for any \(\alpha\in F, \alpha^{|F|}=\alpha\text{.}\)
6. The condition for \(y^2+y+c=0\) to have a root in \(\GF(16)\) is that \(c_3=0\) in our representation. In general, in \(\GF(2^m)\) exactly half of the field yields a quadratic
\begin{equation*}
y^2+y+c
\end{equation*}
with two roots in the field (can see this with the trace function, see problems 3.31 and 3.42 in Roth)
Reed Solomon Codes (Original View).
Let \(F=\GF(2^n)\) and \(\alpha\) be a primitive element of \(F\text{.}\) Map polynomials of degree \(m-1\) in \(F[x]\) to \(F^{2^n}\text{,}\)
\begin{equation*}
p(x) \mapsto (p(0),p(\alpha),p(\alpha^2),\ldots,p(\alpha^{2^n-1}=1))
\end{equation*}
To decode: Pick \(m\) coordinate positions and find the degree \(m-1\) polynomial passing through \((\alpha_i,\beta_i)\) where \((\beta_1,\ldots,\beta_{2^n})\) is the received message. Do this for all subsets of size \(m\) and decode to the plurality computed polynomial.
8. RS code over \(F=\GF(8)\text{,}\) \(m=3\text{.}\)
\begin{equation*}
p(x) =a_0+a_1x+a_2x^2 \mapsto (p(0),p(\alpha),p(\alpha^2),\ldots,p(\alpha^{7}=1))
\end{equation*}
Received \(\by\text{.}\)
Set 12: Find \(a_0+a_1x+a_2x^2=p(x)\) so that
\begin{align*}
p(0) \amp =1\\
p(\alpha) \amp =\alpha\\
p(\alpha^4) \amp =1
\end{align*}
