# Exceed Information limit with a Bigger Codebook? [closed]

I was reading about how there is an upper limit to how fast information can be transmitted over a given channel - the Shannon–Hartley theorem.

I was wondering how this theorem works with codebooks. For example, say my friend and I share a codebook that maps every possible grammatical sentence to an integer.

Then no matter how complex a sentence I want to make, all I need to do is send that single integer! Sure, such a codebook would be huge, huge, huge. But does that matter? I was still able to communicate a sentence of arbitrary complexity just by sending a single number.

Basically, does the Shannon–Hartley theorem take into consideration the size of the codebooks approaching infinity? The codebook itself wouldn't need to be transmitted since we can have a copy sitting with us in advance. Or perhaps do a look up in a database.

If the numbers get very large, my friend and I can work in hexadecimal. Or even a number system with 200 characters. That would cut down the size of the numbers considerably.

I know all this is impractical. I'm just curious about how it works in principle...

## closed as off-topic by CuriousOne, sammy gerbil, Jim, knzhou, ACuriousMind♦Aug 8 '16 at 21:38

• This question does not appear to be about physics within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

• Apart from the difficulty that your codebook could probably not encode Shakespeare, at best you are getting a constant (and relatively small) coding gain that eliminates the inefficiency of the English language. Since English allows to make sentences of infinite length your codebook can never be finite. None of this is physics, though. – CuriousOne Aug 6 '16 at 23:18
• The problem is that your "codes" would be almost as long as the thing you're encoding! Moving to a higher base (hexadecimal) doesn't help because then transmitting each digit is harder. – knzhou Aug 6 '16 at 23:23
• The reason is that bigger cookbooks get you closer to the optimal bound, as you've noticed for English words. So in the infinite limit you get the optimal bound. – knzhou Aug 7 '16 at 5:21
• Whether or not a really big cookbook is practically useful depends on the application. This site isn't the place to ask that question. – knzhou Aug 7 '16 at 5:21
• I'm voting to close this question as off-topic because it is about mathematical theorems in theoretical computer science, not physics – Jim Aug 8 '16 at 20:02