What does it mean to say that $C$ is a binary linear code of length $n$, rank $k$ and minimum distance $d$ ? Let $C$ be such a code.

(a) Prove that $n \geqslant d+k-1$.

Let $x=\left(x_{1}, \ldots, x_{n}\right) \in C$ be a codeword with exactly $d$ non-zero digits.

(b) Prove that puncturing $C$ on the non-zero digits of $x$ produces a code $C^{\prime}$ of length $n-d, \operatorname{rank} k-1$ and minimum distance $d^{\prime}$ for some $d^{\prime} \geqslant\left\lceil\frac{d}{2}\right\rceil$.

(c) Deduce that $n \geqslant d+\sum_{1 \leqslant l \leqslant k-1}\left\lceil\frac{d}{2^{t}}\right\rceil$.