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Here is a non-rhetorical answer to your rhetorical question:

http://www.inference.phy.cam.ac.uk/itila/

See how far you'll get without calculus.

You actually can get pretty far, Huffman coding for example.

If throwing a tome is your style, I suggest going through http://www.cis.upenn.edu/~bcpierce/tapl/ or http://www.decision-procedures.org/ -- you can go through entire books without calculus.

I thought the claim was that you don't need calculus for coding theory, not that there exist branches of advanced mathematics where calculus isn't necessary. So it's germane to produce a book on coding theory and show that the mathematics of the continuum begins literally on page 2.

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