Solution
= Solution
A <linear-feedback shift register> is a linear map
$$
M(x_1,\ldots,x_d)=(x_2,\ldots,x_d,\sum_ia_ix_i).
$$
For the stated matrix $H$, a word $c=(c_0,\ldots,c_{k-1})$ lies in $C$ exactly when $\sum_jc_jM^jx=0$. If $\sigma c$ is its cyclic shift, periodicity $M^kx=x$ gives
$$
H(\sigma c)=M(Hc)=0.
$$
\b[Thus $\ker H$ is shift-invariant and is a binary cyclic code.]