Embedding dimension gaps in sparse codes

TL;DR

This paper investigates the embedding dimensions of a specific 3-sparse code related to the Fano plane, revealing differences between open and closed embedding dimensions and exploring realizability by geometric objects.

Contribution

It provides the first example of a 3-sparse code with closed embedding dimension three and differing open and closed embedding dimensions, and generalizes realizability results for quadratic codes.

Findings

Closed embedding dimension of the code is three.

Open embedding dimension ranges between four and six.

Quadratic codes are realizable by axis-parallel boxes.

Abstract

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.