For $n \in \mathbb{N}$ let $Q_{n}=\{0,1\}^{n}$ denote the set of all $0-1$ sequences of length $n$. We define the distance $d(x, y)$ between two elements $x$ and $y$ of $Q_{n}$ to be the number of coordinates in which they differ. Show that $d(x, z) \leqslant d(x, y)+d(y, z)$ for all $x, y, z \in Q_{n}$.

For $x \in Q_{n}$ and $1 \leqslant j \leqslant n$ let $B(x, j)=\left\{y \in Q_{n}: d(y, x) \leqslant j\right\}$. Show that $|B(x, j)|=\sum_{i=0}^{j}\left(\begin{array}{c}n \\ i\end{array}\right)$.

A subset $C$ of $Q_{n}$ is called a $k$-code if $d(x, y) \geqslant 2 k+1$ for all $x, y \in C$ with $x \neq y$. Let $M(n, k)$ be the maximum possible value of $|C|$ for a $k$-code $C$ in $Q_{n}$. Show that

$$2^{n}\left(\sum_{i=0}^{2 k}\left(\begin{array}{c} n \\ i \end{array}\right)\right)^{-1} \leqslant M(n, k) \leqslant 2^{n}\left(\sum_{i=0}^{k}\left(\begin{array}{l} n \\ i \end{array}\right)\right)^{-1}$$

Find $M(4,1)$, carefully justifying your answer.