Can you help me solve the olympiad problem?

Question

Mathematics

I could not specify any of the tags related to this specialty here. Solved the math.
The operator had set up the terminal to cycle through commands to execute a particular program, but the encoded command dictionary was lost due to the attack. It is urgent to return the drone to the base.
We know what commands the drone executes and what encrypted sequence the terminal transmits. You need to set the command of the shortest length to return to the base and send it to the drone.
There are 6 commands in total: Forward, Back, Right, Left, Signal, Back.
Each of the commands is encrypted with a binary code, no longer than 5 bits.
The entire instruction set satisfies the Fano condition when no instruction code is the beginning of another instruction code.
ENCRYPTED SEQUENCE:
1010111010111001000011100001000111
Commands: Forward Forward Signal Left Forward Signal Left Left Back Signal Back Right Signal

Answer 1

You have already been answered on the site https://ru.stackoverflow.com/questions/1067831/%d0...
And, by the way, why is there one sequence
1010111010111001000011100001000111 on this site
and another
10110111010111001000011100001000111 on the other
?

Answer 2

A scam decision.
The Signal command obviously has code 11 or 111. In both cases code 001 is exactly free in the sequence.
001 - Return (it can be so for sure, nothing prevents this)

Answer 3

Something seems to me that there is an error, or I'm stupid.
Forward = 10 unambiguously, Signal = 111 is obviously readable.
The signal cannot be 11, because otherwise Left would have to be 1 or start with 10 (contradicts Fano in both cases).
From here, Left = 0.
Thus, we reach the middle, but then it turns out that Back = 10000, which contradicts the Fano condition.
Between the last Signals, 00001000 does not fit into the Back + Right combination at all.
===
update. Or does it mean that the bit sequence is cyclically shifted by an unknown number of positions in advance? Well, that is, that this is just a highlighted repeating pattern of radio traffic, and not an exact binding to a sequence of commands?

Answer 4

well
101 - Forward
10000 - Back
10001 - Right
0 - Left
111 -
Back signal -1001
but no, 110 is shorter :)