No "right" answer
In my experience, there are several models I've seen successfully deployed.
The old standard is that coding belongs to HIM (e.g., medical records). Historically, that is where the coding unit reported.
Newer models employ a "divorce", where the coding section actually reports through the CFO. In other words, it becomes part of revenue, not part of HIMS.
Another popular model has coding reporting through the business office or patient accounts. There are actually quite a few facilities I'm familiar with that take this approach.
As for coding reporting through billing, I cannot imagine why this would be done. A.) That does not support separation of duties. (In other words, coders should not be reporting to a billing manager as this is a conflict of interest.) B.) Coding should have an ability to be unbiased and capable of remaining "revenue neutral". C.) A compliant environment supports the two disciplines being separate in that a senior manager cannot be in a position of intimidation to the coding professional.
Now, realize these are just my thoughts.
Kevin B. Shields, RHIT, CPCO, CCS, CPC, COC, CCS-P, CPC-P, CPC-I