The objective is to broadcast information over a noisy digital channel (corruption rate of say, 10-20 %, where bits flip). Assume the broadcasting software is public and there is no possibility of embedding secret information so a public key cryptosystem must be used. Let the adversary have full knowledge of the software but no control or knowledge of a broadcasting node.
Traditional methods to do this such as a "secretbox" or some other DH mechanism will give you a bit-string which is indistinguishabile from random but that string will be subject to corruption if transmitted naked.
So you have to use error correction coding. How to do this so that an adversary listening to the channel cannot detect the presence of the communication by detecting error correction codes? An adversary can do this by continually attempting to decode "noise" over the channel. Decoding tends to fail on random strings.
Note that this only has to be done for the DH part, afterwards an unauthenticated stream cipher can solve this.
From my search it seems that perhaps only Goppa codes in a construction similar to McEliece's Public key cryptosystem can help, but I do not know how to proceed.