Notice that when you analyze the classical binary messages that the mean radial distance increases, but the standard deviation decreases.
It does kind of seem, at first glance, like a "spherization" of the positions in the message (state) space.
The C++ code is attached.
In the following list, the "max message size" is the number of bits per message. So the first in the list analyzes just the two 1-bit messages, the second in the list analyzes the four 2-bit messages, etc, etc. I have to redo the code so that it does not store the radii in an array, since it inevitably runs out of memory on this system when the "max message size" gets to about 25. I can just analyze the radii twice; the first time to get the mean, the second time to get the standard deviation.
The n-bit messages of course "live" in a discrete n-dimensional space that has only two positions (0, 1) for each dimension.
max message size: 1
min radius: 0
max radius: 1
mean radius: 0.5 -/+ 0.5
max message size: 2
min radius: 0
max radius: 1.41421
mean radius: 0.853553 -/+ 0.521005
max message size: 3
min radius: 0
max radius: 1.73205
mean radius: 1.12184 -/+ 0.491409
max message size: 4
min radius: 0
max radius: 2
mean radius: 1.33834 -/+ 0.456989
max message size: 5
min radius: 0
max radius: 2.23607
mean radius: 1.52183 -/+ 0.428974
max message size: 6
min radius: 0
max radius: 2.44949
mean radius: 1.68313 -/+ 0.408759
max message size: 7
min radius: 0
max radius: 2.64575
mean radius: 1.82867 -/+ 0.394921
max message size: 8
min radius: 0
max radius: 2.82843
mean radius: 1.96247 -/+ 0.385622
max message size: 9
min radius: 0
max radius: 3
mean radius: 2.08713 -/+ 0.379348
max message size: 10
min radius: 0
max radius: 3.16228
mean radius: 2.20439 -/+ 0.37503
max message size: 11
min radius: 0
max radius: 3.31662
mean radius: 2.31552 -/+ 0.371968
max message size: 12
min radius: 0
max radius: 3.4641
mean radius: 2.42143 -/+ 0.369719
max message size: 13
min radius: 0
max radius: 3.60555
mean radius: 2.52281 -/+ 0.368006
max message size: 14
min radius: 0
max radius: 3.74166
mean radius: 2.62022 -/+ 0.366657
max message size: 15
min radius: 0
max radius: 3.87298
mean radius: 2.7141 -/+ 0.365562
max message size: 16
min radius: 0
max radius: 4
mean radius: 2.80482 -/+ 0.364652
max message size: 17
min radius: 0
max radius: 4.12311
mean radius: 2.89268 -/+ 0.36388
max message size: 18
min radius: 0
max radius: 4.24264
mean radius: 2.97793 -/+ 0.363215
max message size: 19
min radius: 0
max radius: 4.3589
mean radius: 3.0608 -/+ 0.362634
max message size: 20
min radius: 0
max radius: 4.47214
mean radius: 3.14148 -/+ 0.362121
max message size: 21
min radius: 0
max radius: 4.58258
mean radius: 3.22012 -/+ 0.361665
max message size: 22
min radius: 0
max radius: 4.69042
mean radius: 3.29689 -/+ 0.361256
max message size: 23
min radius: 0
max radius: 4.79583
mean radius: 3.37191 -/+ 0.360886
max message size: 24
min radius: 0
max radius: 4.89898
mean radius: 3.44529 -/+ 0.360552
max message size: 25
min radius: 0
max radius: 5
mean radius: 3.51713 -/+ 0.360246Attachment #1: radius.txt
