MacWilliams identity

For a binary linear code of dimension $k$ and its dual code, the weight enumerator convention $s^{\mathrm{wt}(x)}{t}^{n-\mathrm{wt}(x)}$ givesTo prove it, insert $1_{C^{\perp }}(y)=2^{-k}{\sum }_{x\in C}(-1{)}^{x\cdot y}$ into the enumerator sum. Summing independently over each coordinate of $y$ yields a factor $t+s$ if $x_j=0$ and $t-s$ if $x_j=1$.