Can a noisy encoder be used to communicate reliably?

In this paper the problem of reliable communication with a noisy encoder is examined. We explicitly provided the construction of the encoder and show that even when all logic gates that constitute the encoder are noisy, reliable communication with a positive rate is still possible. The encoding complexity is shown to be O(log 1/p_tar/log1/ε) per bit to achieve a target bit error rate p_tar, where ε denotes the error probability of each noisy gate. This complexity upper bound is shown to coincide with a lower bound in order sense, and is hence tight. The key technique in the proposed construction is to embed noisy decoders inside the noisy encoder, which are utilized repeatedly to prevent the bit error rate from escalating. The proposed noisy encoder has a direct application in noisy computing of a linear transform.