$(12$ points$)$ Suppose that we have a Hamming $(15,2^{11},3)$ code, $C,$ that can be decoded using a method similar to that from Video $6$ of Section $7\mathrm{.}4.$ What would the decoding words be for this code? $($Note that if we generalize the method from Video $6$ of Section $7\mathrm{.}4($which is what I$\text{'}$m asking you to do$),$ then the code associated with these four decoding words cannot be constructed using the method from Theorem $7\mathrm{.}4\mathrm{.}4.)$

Use these decoding words to decode the following words. Justify your choices.

$($a$)110000000000111$

$($b$)000111000010111$

$($c$)111111111101100$