A parity-check matrix is a mathematical representation used in coding theory that helps detect errors in transmitted messages by verifying the parity of codewords. It consists of a matrix where each row represents a linear equation that relates to the bits of a codeword, providing a way to check whether the received codeword is valid or not. This matrix plays a crucial role in error detection and correction techniques, influencing systematic encoding, syndrome decoding, and the construction of various codes.
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