Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 12K / c

If $c\in C$ had odd Hamming weight, add all $n$ cyclic shifts of $c$. Each coordinate of the sum is the sum of all coordinates of $c$, namely one in ${\mathbb{F}}_2$. Linearity and cyclicity would therefore put $(1,\dots ,1)$ in $C$, contrary to the hypothesis. Hence every codeword has even weight.