Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 11I / a / i / Solution

A binary block code of length $n$ is a subset $C\subseteq {\mathbb{F}}_2^n$. Its size is $M=|C|$, and its minimum Hamming distance isThus an $[n,M,d]$ code consists of $M$ binary words of length $n$ whose distinct pairs differ in at least $d$ coordinates. When $d=1$, every set of distinct words qualifies. Taking all of the binary vector space ${\mathbb{F}}_2^n$ gives $2^n$ words, and no code can contain more words than its ambient space. Hence