Question: What is ``the best'' code?

What is the ``best'' code of a given length? This natural, but very hard, question motivates the following definition.

Definition 3.5.1   Let $F$ be a finite field with $q$ elements. Let $A_q(n,d)$ denote the largest $M$ such that there exists a $(n,M,d)$ code in $F^n$.

Determining $A_q(n,d)$ is one of the main problems in the theory of error-correcting codes^3.1. At the time of this writing, $A_2(n,d)$ is known for $n<30$, $d$ arbitrary. The previous example implies that $A_2(7,3)\geq 16$. (It turns out that $A_2(7,3)= 16$.)