Q$4.$ In a $(5,2)$ linear block code, the received codeword $[$v$]$ has five bits and are given as:

$[$r$1$ k$1$ r$2$ k$2$ r$3].$ Here, $k_i$ represents the source code bits and $r_i$ represents the redundant

bits. It is known that $r_1$ and $r_2$ are the LSB and MSB of the source code, respectively

whereas $r_3$ is always $"1".$

$($a$)$ Prepare the list of all the codewords.

$($b$)$ Prove that the codewords in this logic are NOT valid codewords

$($c$)$ Propose a modification to this method so that the codewords will become valid.

Note that while proposing the modification, you are strictly allowed to make only ONE

modification to the logic behind any one redundant bit. It can either be r$1,$ r$2$ or r$3.$

$($d$)$ The $3$rd and $4$th bits of the codeword $[$i$.$e$.,$ v$3$ and v$4]$ are corrupted during

transmission. Verify whether the receiver can detect and as$.$ well as correct these two

errors.

$($e$)$ Determine the efficiency of this method.

$(52=10M)$