Let $C$ be an $[n, m, d]$ code. Define the parameters $n, m$ and $d$. In each of the following cases define the new code and give its parameters.

(i) $C^{+}$is the parity extension of $C$.

(ii) $C^{-}$is the punctured code (assume $n \geqslant 2$ ).

(iii) $\bar{C}$ is the shortened code (assume $n \geqslant 2$ ).

Let $C=\{000,100,010,001,110,101,011,111\}$. Suppose the parity extension of $C$ is transmitted through a binary symmetric channel where $p$ is the probability of a single-bit error in the channel. Calculate the probability that an error in the transmission of a single codeword is not noticed.