Enumeration of left braces with additive group $C_4 \times C_4 \times C_4$

TL;DR

This paper counts the total number of distinct algebraic structures called left braces with a specific additive group of order 64, providing a precise enumeration of 1,515,429 classes.

Contribution

It provides the first exact enumeration of left braces with additive group $C_4 imes C_4 imes C_4$, a significant step in understanding their classification.

Findings

Number of isomorphism classes is 1,515,429

Complete classification for this specific additive group

Advances understanding of algebraic structures related to solutions of the Yang-Baxter equation

Abstract

We show that the number of isomorphism classes of left braces of order 64 with additive group isomorphic to $C_4 \times C_4 \times C_4$ is 1\,515\,429.