Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 11H / a

The parity-check matrix has rank $3$, so its kernel has dimension $7-3=4$ and size $16$; see linear code. Its columns are precisely the seven distinct nonzero vectors of ${\mathbb{F}}_2^3$. Thus no weight-one or weight-two word is in the kernel, while three dependent columns give a weight-three word. Hence $d=3$.