# Minimal Embedding Dimensions of Connected Neural Codes

## TL;DR

This paper characterizes receptive field codes realizable by connected receptive fields and determines that all such codes can be embedded in at most three dimensions, providing the first exact characterization of this family.

## Contribution

It provides a complete characterization of connected receptive field codes and establishes their minimal embedding dimensions, which was previously unknown.

## Key findings

- All connected codes are realizable in dimension at most 3.
- First family of receptive field codes with known exact characterization and minimal embedding dimension.

## Abstract

In the past few years, the study of receptive field codes has been of large interest to mathematicians. Here we give a complete characterization of receptive field codes realizable by connected receptive fields and we give the minimal embedding dimensions of these codes. In particular, we show that all connected codes are realizable in dimension at most 3. To our knowledge, this is the first family of receptive field codes for which the exact characterization and minimal embedding dimension is known.