[turbohedgehog]

The table basically says it's possible to pick a set of 4096 (2^12) different vectors of T/F combinations in a way that you only need flip up to 3 (R=3) of the T/F answers to cover all 2^23 (n=23) possible solutions to the quiz. Thus you only need 68 minutes and 16 seconds (4096 seconds) in order to guarantee you get at least 20 answers correct.

You can use the remaining 1304 seconds to encode some more vectors to increase the possibility of getting more answers correct, but you cannot guarantee 21 or more correct answers in the allotted time. The lower & upper bound (not sure what this part means) for n=23,R=2 is 30686-32768 about 9 hours!

Very simple example with n=3. I have listed all the vectors with Hamming distance of 1 or less (R=1). Looking at the table, we see that for n=3, R=1, we can cover all the vectors with 2 different codes. Heads and tails for example. It's easy to see which combinations we can pick to guarantee coverage, a&h, b&g, c&f, d&e. Using any of those combinations and some way to signal to the other person 'heads' or 'tails' will guarantee they get at least 2 answers correct.

  a TTT <= TTT TTF TFT FTT
  b TTF <= TTT TTF         TFF FTF     
  c TFT <= TTT     TFT     TFF     FFT 
  d FTT <= TTT         FTT     FTF FFT
  e TFF <=     TTF TFT     TFF         FFF
  f FTF <=     TTF     FTT     FTF     FFF
  g FFT <=         TFT FTT         FFT FFF
  h FFF <=                 TFF FTF FFT FFF

[kowdermeister]

This is still an ELI21 at least :)

ELI5-s should use cars cookies or candies to explain :)

[StavrosK]

We always have to be wrong, but we can bunch up all the ways we can be wrong so they're always close to being right. Each way to be right will have a few ways to be wrong as neighbours, so even if we're wrong, we aren't wrong by much.

[kowdermeister]

Interestingly this just came across in my YT feed :)

https://www.youtube.com/watch?v=1_X-7BgHbE0