Binary linear code
= Binary linear code
{title2=$C\leq\mathbb F_2^n$}
A <linear code> over the two-element <finite field>. Its dimension $r$ gives $2^r$ codewords. Its <weight enumerator> in the convention $W_C(s,t)=\sum_{c\in C}s^{w(c)}t^{n-w(c)}$ is symmetric in $s,t$ exactly when the all-ones word belongs to $C$: adding that word complements every bit and replaces weight $w$ by $n-w$. Conversely symmetry makes the coefficient of $s^n$ equal the coefficient of $t^n$, which is one.