(Solved): 4. [25 points] Suppose we have an error-correcting code with the generator matrix shown below. \[ ...

4. [25 points] Suppose we have an error-correcting code with the generator matrix shown below. \[ G=\left[\begin{array}{llllllll} 1 & 0 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 \end{array}\right] \] (a) Is this code systematic? What is the parity check matrix, H? [5] (b) Suppose we wish to send the data word [ \( \left.\begin{array}{llll}1 & 1 & 1 & 1\end{array}\right] \), what is the transmitted codeword? [5] (c) If we have an error in the first bit when transmitting the codeword in part b, what is the output of the decoder? How do we determine and correct the error? [5] (d) If we have an error in the \( 5^{\text {th }} \) and \( 6^{\text {th }} \) bits, what is the output of the decoder? What does this tell you about this code's error-correcting capability? [10]