RU-SPB 2021 Problem 104

Stierlitz wants to send an encryption to the Center, which is a code containing $100$ characters, each a "dot" or a "dash". The instruction he received from the Center the day before about conspiracy reads:

i) when transmitting encryption over the radio, exactly $49$ characters should be replaced with their opposites;

ii) the location of the "wrong" characters is decided by the transmitting side and the Center is not informed of it.

Prove that Stierlitz can send $10$ encryptions, each time choosing some $49$ characters to flip, such that when the Center receives these $10$ ciphers, it may unambiguously restore the original code.