Lets say I have the expression $A_q (n,d)=y$.
I understand that y would be the maximum value for M for which there exists a q-ary $(n,M,d)$.
What does q-ary mean? Is it just he number of elements?
Lets say I am the arbitrary made up example of $A_3 (3,2)=5$
What does the 3 mean?
How could I show by construction that this equality holds?
What would the general outline be?
A code is $q$-ary if the number of elements in the alphabet is $q$.
To show that $A_{3}(3,2) = 5$, you would need to construct a code with alphabet of size 3, length of codewords 3, minimum distance 2, having 5 codewords. Then you would need to show that there does not exist such a code with more than 5 codewords (through an exhaustive search, or application of a known upper bound on the size of a code).