MathDB
Stierlitz encrypts a 100-digit binary string

Source: St Petersburg 2021 10.4

December 23, 2021
combinatorics

Problem Statement

Stierlitz wants to send an encryption to the Center, which is a code containing 100100 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 4949 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 1010 encryptions, each time choosing some 4949 characters to flip, such that when the Center receives these 1010 ciphers, it may unambiguously restore the original code.