Problem $3$

Another simple code is the binary Hamming code of block length $7.$ Here there are four information bits, $A_1,A_2,A_3,A_4$ and three parity checks $A_5,A_6,A_7.$ The parity checks are chosen as follows:

$A_5=A_{1}o+A_{2}o+A_3$

$A_6=A_{2}o+A_{3}o+A_4$

$A_7=A_{1}o+A_{2}o+A_4$

In these equations, $o+$ stands for modulo $2$ addition. Hence, the parity check has value $1$ if the number of $1\text{'}$s in the set being checked is odd and has value $0$ otherwise. Similar to Problem $2,$ the binary digits are mapped into antipodal signals so that $E_s=$ for the symbol associated with each codeword. WGN of spectral density $\frac{N_0}{2}$ is added and the received waveforms are matched filtered to produce $Y_1,dots{Y}_7,$ where, given any information sequence, the $Y_i$ are independent with mean $\sqrt[\phantom{2}]{}$ and variance $\frac{N_0}{2}.$

$($a$)$ Write out all $16$ codewords. Find the number of codewords with weight $3,4,$ and $7.$

$($b$)$ Assuming that the codewords are equally likely and the maximum likelihood detection is used, use the union bound to upper bound the probability of error given that the all$-$zero codeword is transmitted.

$($c$)$ Explain why the probability of error is the same for each other codeword. Hint: Given any codeword $c,$ add that codeword to any other codeword $c^{\text{'}}.$ Argue that $co+c^{\text{'}}$ is also a codeword and show that the Hamming distance between $c$ and $c^{\text{'}}$ is the same as the distance between the all$-$zero codeword and $co+c^{\text{'}}.$

$($d$)$ Evaluate the effective coding gain for this code.