Consider the binary code C $=\{$c$1=10101,$c$2=01110,$c$3=11001\}.$ Suppose that P$($c$1)=0\mathrm{.}5,$ P$($c$2)=0\mathrm{.}2,$ and P$($c$3)=0\mathrm{.}3,$ where P$($ci$)$ denotes the probability that ci is sent. Suppose that a binary symmetric channel with symbol error probability p is being used, and r $=00110$ is received.

$($a$)$ What is the distance of C$?$

$($b$)$ Suppose that p $=0\mathrm{.}1.$ Decode r using IMLD.