Embedding dimension gaps in sparse codes

We study the open and closed embedding dimensions of a convex 3-sparse code $\mathcal{FP}$, which records the intersection pattern of lines in the Fano plane. We show that the closed embedding dimension of $\mathcal{FP}$ is three, and the open embedding dimension is between four and six, providing the first example of a 3-sparse code with closed embedding dimension three and differing open and closed embedding dimensions. We also investigate codes whose canonical form is quadratic, i.e. ``degree two" codes. We show that such codes are realizable by axis-parallel boxes, generalizing a recent result of Zhou on inductively pierced codes. We pose several open questions regarding sparse and low-degree codes. In particular, we conjecture that the open embedding dimension of certain 3-sparse codes derived from Steiner triple systems grows to infinity.