A subset $\mathcal{C}$ of the Hamming space $\{0,1\}^{n}$ of cardinality $|\mathcal{C}|=r$ and with the minimal (Hamming) distance $\min \left[d\left(x, x^{\prime}\right): x, x^{\prime} \in \mathcal{C}, x \neq x^{\prime}\right]=\delta$ is called an $[n, r, \delta]$-code (not necessarily linear). An $[n, r, \delta]$-code is called maximal if it is not contained in any $[n, r+1, \delta]$-code. Prove that an $[n, r, \delta]$-code is maximal if and only if for any $y \in\{0,1\}^{n}$ there exists $x \in \mathcal{C}$ such that $d(x, y)<\delta$. Conclude that if there are $\delta$ or more changes made in a codeword then the new word is closer to some other codeword than to the original one.

Suppose that a maximal $[n, r, \delta]$-code is used for transmitting information via a binary memoryless channel with the error probability $p$, and the receiver uses the maximum likelihood decoder. Prove that the probability of erroneous decoding, $\pi_{\mathrm{err}}^{\mathrm{ml}}$, obeys the bounds

$$1-b(n, \delta-1) \leqslant \pi_{\mathrm{err}}^{\mathrm{ml}} \leqslant 1-b(n,[(\delta-1) / 2])$$

where

$$b(n, m)=\sum_{0 \leqslant k \leqslant m}\left(\begin{array}{l} n \\ k \end{array}\right) p^{k}(1-p)^{n-k}$$

is a partial binomial sum and $[\cdot]$ is the integer part.