> [!tldr] Introduction > > If a communication channel is noisy, then the sent codeword is not always the same as the received codeword, meaning that there needs to be a rule that determines what codeword was sent based on the codeword that was received. > [!definition] > > Let $C = \list{x}{N}$ be the [[Set|set]] of all possible [[Code|code]]words that can be transmitted across a [[Communication Channel|communication channel]], and let $y$ be the codeword received. A decision rule is a procedure that systematically assign a unique transmitted codeword to any one received.