(Solved): 2. Hamming 8,4 G=11100001100110010101010111010010H=1001010111010011 ...

2. Hamming 8,4 $$\mathrm{G}=\left(\begin{array}{llll}1& 1& 0& 1\\ 1& 0& 1& 1\\ 1& 0& 0& 0\\ 0& 1& 1& 1\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 1& 1& 0\end{array}\right)\mathrm{H}=\left(\begin{array}{llllllll}1& 0& 1& 0& 1& 0& 1& 0\\ 0& 1& 1& 0& 0& 1& 1& 0\\ 0& 0& 0& 1& 1& 1& 1& 0\\ 1& 1& 1& 1& 1& 1& 1& 1\end{array}\right)\mathrm{R}=\left(\begin{array}{llllllll}0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\end{array}\right)$$ Given : - Code generator matrix $$G$$ - Parity check matrix $$\mathrm{H}$$ - Decoder matrix $$\mathrm{R}$$ a) For 4 bit word 0011 , show the transmitted codeword (8 bit). b) Assume that the same codeword you have designed in a) has been transmitted and received with no error. Show how to check the received word using the parity check matrix and decode to show that there has been no error during transmission. c) Assume that the codeword you have designed in a) was transmitted, but upon being received the $$4^{\text{th }}$$ bit of the codeword has been flipped due to noise. Show how the parity check matrix can detect the error and fix it. Decode the code after the correction to prove that the transmitted code is correct. Show all steps. d) What happens if in part c) $$3^{\text{rd }}$$ bit and the $$4^{\text{th }}$$ bit have been flipped due to noise? Can we detect the error? Can we recover the transmitted codeword? Explain in detail.