A **fusion ring** $\mathcal{F}$ (see [here](http://www.imsc.res.in/~sunder/pf.pdf) p 28 for more details) is **integral** if the Perron-Frobenius dimension $d(h_i)$ of its basic elements $\{h_1,...,h_r\}$, are integers.   Its **rank** is $r$ and its **dimension** is $\sum d_i^2$.  
   $\mathcal{F}$ is **simple** if it has no non-trivial fusion subring.  
    
The Grothendieck ring of a fusion category (see [here](http://arxiv.org/abs/math/0203060) for more details) is a fusion ring.    
$\mathcal{F}$ is **categorifiable** if it is the Grothendieck ring of a fusion category.  

Let $Rep(G)$ be the fusion category of representations of a finite group $G$.  
The Grothendieck ring $\mathcal{F}(G)$ of $Rep(G)$, is simple iff $G$ is a simple group. 

For all ranks $<9$ and all dimensions $<210$, all the integral simple fusion rings are **trivial** (i.e. isomorphic to a $\mathcal{F}(G)$),
and so are categorifiable. 

> **Question** : Is an integral simple fusion ring, always categorifiable ?

The first non-trivial simple integral fusion ring I know, is rank $7$ and dimension $210$ (see [here](http://mathoverflow.net/questions/132866/non-weakly-group-theoretical-integral-fusion-categories)).