Quiz

And please, don't use Wikipedia or search engines!

Yes, we вроде agreed on this at the beginning of the quiz.

In general, I looked at what was suggested here and propose not to transmit the original message at all. Or rather, like this.
Let the original message consist of 4 bits: i1, i2, i3, i4
Then we transmit the following:
bit1: i1
bit2: i1
bit3: i2
bit4: xor(i1, i2)
bit5: xor(bit4, i3)
bit6: xor(bit5, i4)
bit7: xor(i3, i4)

I haven't checked all cases, but at first glance, there is a double check everywhere, and it is possible to determine which bit is faulty and its actual value.

It's just that the example turned out to be not very simple. Perhaps someone will find a simpler solution.

Added 7 minutes later
xor (0,0) = 0
xor (0,1) = 1
xor (1,0) = 1
xor (1,1) = 0

i.e., this checks whether the operands are the same or different.

Example.
It is necessary to transmit 1 1 1 1
We transmit 1 1 1 0 1 0 0