Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 12I / iii / Solution

Adding the all-one codeword is a bijection from $C$ to itself, and it replaces every bit by its complement. It therefore sends weight $j$ to weight $n-j$, provingThe code has one word of weight zero and, since its minimum Hamming distance is three, no words of weight one or two. Combining this with part (i) gives