In this chapter, we introduce the model of a communication system, as originally proposed by Claude E. Shannon in 1948. We will then focus on the channel portion of the system and define the concept of a probabilistic channel, along with models of an encoder and a decoder for the channel. As our primary example of a probabilistic channel—here, as well as in subsequent chapters—we will introduce the memoryless q-ary symmetric channel, with the binary case as the prevailing instance used in many practical applications. For q = 2 (the binary case), we quote two key results in information theory. The first result is a coding theorem, which states that information through the channel can be transmitted with an arbitrarily small probability of decoding error, as long as the transmission rate is below a quantity referred to as the capacity of the channel. The second result is a converse coding theorem, which states that operating at rates above the capacity necessarily implies unreliable transmission.

In the remaining part of the chapter, we shift to a combinatorial setting and characterize error events that can occur in channels such as the q-ary symmetric channel, and can always be corrected by suitably selected encoders and decoders. We exhibit the trade-off between error correction and error detection: while an error-detecting decoder provides less information to the receiver, it allows us to handle twice as many errors.