Definition of a Perfect Code in Coding Theory

An $(n,M,d{)}_q$-code over the alphabet $A$ is perfect if the following holds:

$$M=\frac{q^n}{\text{vol}(S_{ϵ(d)}(c))}:ϵ(d)=\lfloor \frac{d-1}{2}\rfloor ,\text{for some codeword }c\in C$$

The trivial solution to this is called the “easy” code $C=A^n$.

In other words, a type of q-ary code whose parameters satisfy the Hamming bound with equality.