# Neural Codes and the Factor Complex

## TL;DR

This paper introduces the factor complex of a neural code, providing algebraic and combinatorial characterizations of certain classes of codes, enhancing understanding of their structure.

## Contribution

It presents the novel concept of the factor complex and uses it to characterize max-intersection-complete and intersection-complete neural codes.

## Key findings

- Factor complex captures intervals and maximal codewords.
- Provides algebraic characterization of max-intersection-complete codes.
- Offers a new combinatorial characterization of intersection-complete codes.

## Abstract

We introduce the factor complex of a neural code, and show how intervals and maximal codewords are captured by the combinatorics of factor complexes. We use these results to obtain algebraic and combinatorial characterizations of max-intersection-complete codes, as well as a new combinatorial characterization of intersection-complete codes.