Consider a discrete time channel with binary inputs and binary outputs. Let $x_{n}$ denote the input bit at time $n$ and $y_{k}$ denote the output bit at time $\text{k}$. The channel operation is such that to produce the output $y_{n}$ it drops one of the two inputs $x_{2 n}, x_{2 n+1}$ and outputs the other. Thus $y_{n}=x_{2 n}$ with probability $1 / 2$ and $y_{n}=x_{2 n+1}$ with probability $1 / 2$. Suppose we wish to send $\text{M}$ messages using length $2 N$ inputs. Let $R=\log _{2}(M) / 2 N$. Which of the following is true.
- If $R>1 / 2$, then we always make errors
- If $R=1 / 2$, then there is a transmission scheme such that we do not make any error
- If $R<1 / 2$, then there exists a scheme with zero error
- All of the above
- None of the above