Let C be the two$-$error correcting Reed$-$Solomon code that results from the primitive polynomial $$in$.$

$$ a$.$ Construct a polynomial codeword in C and then convert this polynomial into a binary vector. $(6$ pts$)$

$$ b$.$ How many codewords does this code contain? $(1$ pt$)$

$$c$.$ Provided that only one error burst occurs during transmission of the binary equivalent of a polynomial codeword in C $,$ what is the error burst length that we would be guaranteed to be able to correct? In a best case scenario, how many errors can the code correct? $(2$ pts$)$

$$ d$.$ Correct the following received polynomials. $(14$ pts$)$

$$ i$).$

$.$

$$ ii$)..$

$$

$.$

$$

Hint: Start by decomposing m and n into an arbitrary product of prime factors. What does the $$tell you about this decomposition? Then use Exercise $7$ in Section $13$ is probably easier to start with the right$-$hand side first. $(6$ pts$)$

t

$.(6$ pts