Is there a code that is Maximum Distance Separable and not isomorphic to Reed Solomon Codes? When is a MDS code isomorphic to Reed Solomon Code? 


Is there an easy test? If so, could someone provide me a reference?


In short, if given a $[n,k,n-k+1]\_q$ code how do you show it is RS or not?



On MDS code property
I guess you exclude trivial MDS codes, generalized Reed-Solomon codes, and MDS codes that can be obtained by code extension.


If you exclude them all, there are still a bunch of MDS codes. In general, MDS codes of length $n$ and dimension $k$ over $\mathbb{F}\_q$ are equivalent to $n$-arcs in $\text{PG}(k-1,q)$. Generalized Reed-Solomon codes are $n$-arcs which are subsets of normal rational curves. So, other $n$-arcs are not Reed-Solomon codes. A quick google search gave me the following paper that enumerates all nonequivalent MDS codes of dimension $2$ and $3$ over $\mathbb{F}\_q$ for $4 \leq q \leq 32$:


[G. Kéri, Classification of $2$ and $3$ dimensional MDS codes for $4 \leq q \leq 32$.](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.58.7429)


The equivalence between MDS codes and arcs in $\text{PG}$ is well-known. I think you can find a reference that gives the proof somewhere with a little bit of searching.


