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Go through all of the necessary steps to build a generator polynomial for a 3-error correcting 11-ary Reed-Solomon code of length $10$.

How to go about this? Based on the formula, $(x−βl+j)$, what could I make $β$? Since it has to be a root of an irreducible polynomial, but I am confused on how to incorporate all this information in this problem. Please help in how to start this problem?

Shevliaskovic
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eek
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  • As it happens in this answer I constructed a generator polynomial for a 2-error correcting 11-ary Reed-Solomon code. You can modify the construction to make it 3-error correcting. You get a degree six polynomial (as opposed to a degree four polynomial I arrived at). The recipe is (no surprise) similar to the one described in Intredasting's +1 answer. – Jyrki Lahtonen May 07 '14 at 13:57

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You can actually use an easy formula mentioned earlier in the Spence text (which I assume you're using). Namely, $g(x) = (x - \alpha^b)(x-\alpha^{b+1})...(x-\alpha^{b + \delta - 2})$, where $\delta$ is the desired Hamming distance (in this case, it's $2 * 3 + 1 = 7$), $b$ is any integer $\geq 0$ ($0$ seems like a good choice), and $\alpha$ is a primitive root modulo $11$; for example, $2$.

Brandon
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