3 ms·

All sequences of 5 numbers in the range [6] should be 6 * 6 * ... * 6 = 6^5, which would give 12.92 bits, so 13. The only way to actually do better than sending

All sequences of 5 numbers in the range [6] should be 6 * 6 * ... * 6 = 6^5, which would give 12.92 bits, so 13. The only way to actually do better than sending every symbol would be run length encoding or other non fixed alphabet encodings, which fortunately do not need to be longer than the entropy of the distribution.

nico 3y ago
You are right, it’s 6^5 to include sequences with repeating symbols In any case, you don’t need to index the whole numbers space to transmit all the information Like you say, you can use 13 bits instead of 15 And using variable length encoding you can reduce it even more But what is the distribution in this case?
- rododendro 3y ago
  Well in this case the distribution would simply be a uniform distribution over [1:6], which has entropy 2.58 for each symbol. Which is also the most entropic distribution on the sets of size 6. Like other have said, you can't actually know in a real world scenario your underlying distribution in whatever file you are trying to compress, but since you want to compress it you are telling the program that the file has interesting information (otherwise it would be just noise) and so grammar based encoding would probably not be best, instead lz77 or lz78 would perform better (for example, if you know more about the file you could even do better by transforming first into some frequency domain and then compressing).
  - nico 3y ago
    Fascinating. Thank you. Why do you need to know more about the file to transform to frequency domain?
    - rododendro 3y ago
      Because no information is lost if you simply change domain, but you may spot more patterns faster or easier. For example (even if a bit dated) https://ieeexplore.ieee.org/abstract/document/1614066 https://ieeexplore.ieee.org/abstract/document/1614066 There is a lot of research in this field because of the IoT fad/trend. If you want to know more i would suggest to simply pic a paper from IEEE and deep dive into the terms and terminology and simply skip the proofs of most algorithms/theorems.
contravariant 3y ago
You don't need variable length you just need to send the data in bigger blocks. Count all 6^5n ways to throw 5 dice n times, order them lexicographically and send the index in 5nlog(6) bits, rounding up. Your inefficiency is just the rounding error, which quickly becomes trivial. In theory some block sizes should even coincidentally be close to a whole number of bits.

nico 3y ago

You are right, it’s 6^5 to include sequences with repeating symbols In any case, you don’t need to index the whole numbers space to transmit all the information Like you say, you can use 13 bits instead of 15 And using variable length encoding you can reduce it even more But what is the distribution in this case?

rododendro 3y ago
Well in this case the distribution would simply be a uniform distribution over [1:6], which has entropy 2.58 for each symbol. Which is also the most entropic distribution on the sets of size 6. Like other have said, you can't actually know in a real world scenario your underlying distribution in whatever file you are trying to compress, but since you want to compress it you are telling the program that the file has interesting information (otherwise it would be just noise) and so grammar based encoding would probably not be best, instead lz77 or lz78 would perform better (for example, if you know more about the file you could even do better by transforming first into some frequency domain and then compressing).
- nico 3y ago
  Fascinating. Thank you. Why do you need to know more about the file to transform to frequency domain?
  - rododendro 3y ago
    Because no information is lost if you simply change domain, but you may spot more patterns faster or easier. For example (even if a bit dated) https://ieeexplore.ieee.org/abstract/document/1614066 https://ieeexplore.ieee.org/abstract/document/1614066 There is a lot of research in this field because of the IoT fad/trend. If you want to know more i would suggest to simply pic a paper from IEEE and deep dive into the terms and terminology and simply skip the proofs of most algorithms/theorems.

contravariant 3y ago

You don't need variable length you just need to send the data in bigger blocks. Count all 6^5n ways to throw 5 dice n times, order them lexicographically and send the index in 5nlog(6) bits, rounding up. Your inefficiency is just the rounding error, which quickly becomes trivial. In theory some block sizes should even coincidentally be close to a whole number of bits.