Even-weight subcode of a binary linear code
= Even-weight subcode of a binary linear code
{title2=$C^{\mathrm{ev}}=\ker(\sigma|_C)$}
For a <binary linear code> $C$, the <linear functional> $\sigma(x)=\sum_jx_j$ records <Hamming weight> modulo two. Its <kernel of a linear map> is the even-weight subcode. If $\sigma|_C$ vanishes, this is all of $C$; otherwise <rank-nullity theorem> gives <dimension> $\dim C-1$. This also proves closure under addition without treating ordinary integer weight as a linear function.