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Embedding
dimension gaps in sparse codes
R. Amzi Jeffs, Henry Siegel, David Staudinger and Yiqing Wang |
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Vol. 19 (2026), No. 3, 405–422
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Abstract |
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Every collection of convex sets can be associated to a combinatorial code recording the regions that the sets cut out in space, analogous to a Venn diagram. These “convex codes” have been studied as a mathematical model of hippocampal place cells, and a central question is to find criteria for bounding the minimal “embedding dimension” in which a given code can be realized. We study the open and closed embedding dimensions of a convex 3-sparse code , which records the intersection pattern of lines in the Fano plane. We show that the closed embedding dimension of 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-” 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. |
Keywords
convex, code, sparse, embedding dimension, intersection
pattern, $d$-representable
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Mathematical Subject Classification
Primary: 52A20, 52C99
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Milestones
Received: 7 June 2024
Revised: 8 September 2024
Accepted: 9 September 2024
Published: 12 June 2026
Communicated by Steven J. Miller
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Authors |
| R. Amzi Jeffs | |
| Department of Mathematical
Sciences Carnegie Mellon University Pittsburgh, PA United States |
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| Henry Siegel | |
| Department of Mathematical
Sciences Carnegie Mellon University Pittsburgh, PA United States |
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| David Staudinger | |
| Department of Mathematical
Sciences Carnegie Mellon University Pittsburgh, PA United States |
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| Yiqing Wang | |
| Department of Mathematics University of Wisconsin-Madison Madison, WI United States |
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| © 2026 MSP (Mathematical Sciences Publishers). |
