[[Linear code]]
# Projective system of a linear code

Let $\mathcal{C} \leq \mathbb{K}_{q}^n$ be a [[linear code]], and $c_{i} : \mathcal{C} \to \mathbb{K}_{q}$ denote the $i$th coördinate function,
and let $[c_{i}]$ denote the corresponding point in the [[projectivization]] $\opn P(\mathcal{C}^*) \cong \opn{PG}(k-1,q)$.
The [[multiset]] $\mathfrak{P}(\mathcal{C}) = \mathcal{M}([c_{i}])_{i=1}^n \sube \opn P(\mathcal{C}^*)$ is then called the **projective system** of $\mathcal{C}$. #m/def/code
Iff all elements of $\mathfrak{P}(\mathcal{C})$ occur exactly once, then $\mathcal{C}$ is a [[projective code]].[^2017]

  [^2017]: 2017\. [[Sources/@kwiatkowskiGraphsProjectiveCodes2017|The graphs of projective codes]], §2.2, p. 3

## Properties

- The sum of $\mathfrak{P}(\mathcal{C})$ is $\mathcal{C}^*$.
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