$11\mathrm{.}0.$ In the text it is claimed $$there is no N$-$bit error code that catches all N$-$bit errors$$ for N$>=2($for N$=1,$ a parity bit works$).$ Prove this claim for N$=2.$ Hint: pick a length M$,$ and consider all M$-$bit messages with a single $1-$bit. Any such message can be converted to any other with a $2-$bit error. Show, using the Pigeonhole Principle, that for large enough M two messages m$1$ and m$2$ must have the same error code, that is$,$ e$($m$1)=$ e$($m$2).$ If this occurs, then the error code fails to detect the error that converted m$1$ into m$2.$