A *fusion ring* is a finite dimensional complex space $\mathbb{C}\mathcal{G}$ together with a distinguished basis $\mathcal{G} = \{ h_1,...,h_r\}$ and fusion rules $ h_i \cdot h_j = \sum_k n_{ij}^kh_k $, with $n_{ij}^k \in \mathbb{N}_{\geq 0}$ satisfying:  
 - *Neutral*: $n_{1i}^j = n_{i1}^j = \delta_{ij}$        
 - *Dual*: $\forall i \  \exists!j $ (noted $i^*$) such that $n_{ij}^1>0$  
 - *Associativity*: $\sum_s n_{ij}^sn_{sk}^t = \sum_s n_{jk}^sn_{is}^t$   
 - *Frobenius-Perron reciprocity*: $n_{ij}^k = n_{i^*k}^j = n_{kj^*}^i$    
 
*Remark*: $\mathbb{C}\mathcal{G}$ admits a structure of complex involutive algebra (take $h_i^* = h_{i^*}$).  
*Frobenius-Perron theorem*: there is a unique $*$-homomorphism $d:\mathbb{C}\mathcal{G} \to \mathbb{C}$ with $d(\mathcal{G}) = (0,\infty)$.

The fusion ring $\mathbb{C}\mathcal{G}$ is called *integral* if every $d(h_i)$ is an integer. Its *rank* is the cardinal of $\mathcal{G}$, and its Frobenius-Perron dimension (FPdim) is $\sum d(h_i)^2$.  It is *simple* if for any fusion subring $\mathbb{C}\mathcal{S} \subseteq \mathbb{C}\mathcal{G}$ with $\mathcal{S} \subseteq \mathcal{G}$, we have $\mathcal{S} = \{ h_1 \}$ or $\mathcal{G}$.  

 We have checked by SAGE (using [this code][1]) that the only integral simple fusion ring of rank $\leq 5$ and FPdim $\leq 11500$ is the Grothendieck ring of the simple group $A_5$. It is of rank $5$ and FPdim $60$.

**Question**: Is there an integral simple fusion ring of rank $ \leq 5$ and FPdim $>60$?    
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*Digression*: a fusion ring is called non-trivial if it is not the Grothendieck ring of a finite group. The first non-trivial integral simple fusion ring found by SAGE is of rank $7$ and FPdim $210$ (see [here][2]).


  [1]: https://drive.google.com/file/d/0B2P_JgZe-Zd0RG53WGdmZlFyRzQ/view?usp=sharing
  [2]: http://mathoverflow.net/q/132866/34538