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

A linear-feedback shift register is a linear mapFor the stated matrix $H$, a word $c=(c_0,\dots ,c_{k-1})$ lies in $C$ exactly when ${\sum }_j{c}_j{M}^{j}x=0$. If $\sigma c$ is its cyclic shift, periodicity $M^{k}x=x$ givesThus $\ker H$ is shift-invariant and is a binary cyclic code.

Solved by gpt-5.6-sol high.