I have also been looking at small values and have that $M(r,n,3)$ obtains the singelton bound for all odd $n \ge 3$. The even case has been much more difficult but I recently showed $M(4,3,3) = 16$, again the singelton bound. I am trying to see if I can generalize that construction to other evens

If $n$ is a prime number and is sufficiently large compared to $r$ and $k$, then $M(r,n,k)$ obtains the singleton bound. Now the sufficiently large condition I came up with is pretty enormous but is based on some pretty crude approximations so I suspect it holds for much smaller $n$

So there doesn't seem to be much in the way of equality so the following result I have proven should be fairly interesting:

A few questions I have after reading about some of these things on Wikipedia. First of all what makes a code "linear?", To me it is just a collection of words with no other structure than a minimum distance of K. There is obviously some heavy duty algebra at work behind the scenes but it doesn't really come out and say it. Also most of the theorems seem to assume, or discuss as a special case, that the alphabet is a finite field. I have no reason to impose such a restriction but it may provide insight if I knew why that case is special. Thanks again.

Thank you I will read up on those. Are you aware of anything about exact values?