[[Weight enumerator]]
# MacWilliams identities

Let $\mathcal{C} \leq \mathbb{K}_{q}^n$ be a [[Linear code|$[n,k]$ code]] and $\mathcal{C}^\perp$ be its $[n, n-k]$ [[Orthogonal code]].
Let $A(z) = \wt(\mathcal{C}; z)$ and $B(z) = \wt(\mathcal{C}^\perp; z)$ be their respective [[Weight enumerator|weight enumerators]]. Then[^1999] #m/thm/code 
$$
\begin{align*}
B(z) = q^{-1}(1+ (q-1)z^n) A\left( \frac{1-z}{1+(q-1)z} \right)
\end{align*}
$$

  [^1999]: 1999\. [[Sources/@vanlintIntroductionCodingTheory1999|Introduction to coding theory]], §3.5, pp. 41–42

> [!missing]- Proof
> #missing/proof

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