[[Linear code]]
# Projective code

A $[k,n]-$[[linear code]] $\mathcal{C} \leq \mathbb{K}_{q}^n$ is said to be **projective** iff it meets the following equivalent conditions:[^1999][^2017] #m/def/code 

- its [[Orthogonal code]] $\mathcal{C}^\perp$ has [[Code#^minimumDistance]] $d \geq 3$;
- any [[Linear code#^generator]] $G$ of $\mathcal{C}$ has pairwise [[Linear (in)dependence|linearly independent]] columns; or
- every point in the [[Projective system of a linear code|projective system]] $\mathfrak{P}(\mathcal{C})$ occurs exactly once.


  [^1999]: 1999\. [[Sources/@vanlintIntroductionCodingTheory1999|Introduction to coding theory]], §3.3, p. 38
  [^2017]: 2017\. [[Sources/@kwiatkowskiGraphsProjectiveCodes2017|The graphs of projective codes]], §2.2, p. 3
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