Exact Minimum Density of Codes Identifying Vertices in the Square Grid

An identifying code C is a subset of the vertices of the square grid ${\mathbb Z}^2$ with the property that for each element v of ${\mathbb Z}^2$, the collection of elements from C at a distance of at most one from v is nonempty and distinct from the collection of any other vertex. We prove that the minimum density of C within ${\mathbb Z}^2$ is $\frac{7}{20}$.

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