[[Coding theory MOC]]
# Weight enumerator

Let $\mathcal{C} \sube S_{q}^n$ be a $q$-ary [[code]] of length $n$.
Then the **weight enumerator** is of $\mathcal{C}$ is defined by[^1988][^1999] #m/def/code
$$
\begin{align*}
\wt(\mathcal{C};q) = \sum_{c \in \mathcal{C}} q^{\wt c} \in \mathbb{Z}[q]
\end{align*}
$$


  [^1988]: 1988\. [[Sources/@frenkelVertexOperatorAlgebras1988|Vertex operator algebras and the Monster]], §10.1, p. 300
  [^1999]: 1999\. [[Sources/@vanlintIntroductionCodingTheory1999|Introduction to coding theory]], §3.5, pp. 40–41

## Properties

- [[MacWilliams identities]] relate the weight enumerator of a code and its dual.

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