Lecture 3: Coding Theory 1.1 Noisy Communication 1.2 Equality Testing

Alice gets a k-bit message x to send to Bob. Unfortunately, the communication channel is corrupted, and some number of bits may flip. For conceretness, say twenty percent of all bits sent on the channel may flip. Which bits are flipped is unknown, and different bits may flip for different messages sent on the channel. Can Alice still send a message that can be reliably received by Bob? We allow Alice and Bob to make any agreements prior to Alice learning about her message x. We also allow Alice to send more than k bits on the channel. The redundant bits can compesate for the bits lost due to corruption. We will see that Alice can send only n = Θ(k) bits of information about her message, such that even if twenty percent of those bits are flipped, the mistakes can be detected, and all of the original k bits can be recovered by Bob!