Q:

Bounds on error correcting codes

So I was reading 'Introduction to Coding Theory' by van Lint and in the chapter regarding bounds on codes, he has defined a quantity $A(n,d)$=max {$M$ |an $(n,M,d)$ code exists} ( $M$  is the number of codewords) and all the calculations are done with this quantity. My question is, why not do it on $M$ directly instead?

A:

The basic goal of designing a code is efficiency, so given $n,d$ the basic goal is to maximize $M$ so we can send as many messages as possible for a given performance level defined by a given blocklength and a given minimum distance.
Also, extremal codes give rise to some beautiful algebra and combinatorics, e.g., the Golay codes.

