1.
Show that when a GRS code \(\GRS_{n,k}(\balpha,\bv)\) is primitive, then a set of column multipliers of its dual code are \(u_i=-\alpha_i/v_i\text{.}\)
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Worksheet GRS and RS Codes
Today we’ll look at generalized Reed-Solomon (GRS) and Reed-Solomon (RS) codes in more detail. We’ll see that their duals are also GRS/RS codes respectively, and talk about a different encoding perspective from the one we discussed in the previous activity.
2.
Last time we considered the (singly-extended) \(\RS_{8,3}\) code over \(F=\GF(8)\) with \(p(x)\in F_3[x]\mapsto (p(0),p(\alpha),\ldots,p(\alpha^7)) \text{.}\) Give a set of code locators and column multipliers for the non-extended version of this code, i.e.
\begin{equation*}
p(x) \mapsto (p(1),p(\alpha), p(\alpha^2),\ldots, p(\alpha^6))\text{.}
\end{equation*}
3.
Give the canonical generator and parity check matrices for the code in the last exercise.
4.
Is this code primitive? Narrow-sense? Normalized?
5.
Compute the generator polynomial \(g(x)\) for the code in the last exercise.
6.
7.
Encode the message polynomial \(m(x)=1+\alpha x^2\) by remaindering:
\begin{equation*}
m(x)\mapsto x^{n-k}m(x)-r_m(x)
\end{equation*}
where \(r_m(x)\) is the remainder on division of \(x^{n-k}m(x)\) by \(g(x)\text{.}\)
8.
What do you notice about the second encoding?
| Power of \(\alpha\) | Field Element | Coordinates |
|---|---|---|
| \(-\infty\) | \(0\) | 000 |
| \(\alpha^0\) | \(1\) | 100 |
| \(\alpha^1\) | \(\alpha\) | 010 |
| \(\alpha^2\) | \(\alpha^2\) | 001 |
| \(\alpha^3\) | \(1+\alpha \) | 110 |
| \(\alpha^4\) | \(\alpha + \alpha^2\) | 011 |
| \(\alpha^5\) | \(1 + \alpha + \alpha^2\) | 111 |
| \(\alpha^6\) | \(1+\alpha^2 \) | 101 |
