Fig: Error-control taxonomy: detection (parity, checksum, CRC) versus correction, which splits into forward error correction (block and convolutional) and ARQ retransmission schemesFig: CRC flow: the message is shifted by x^r, divided mod-2 by the generator g(x), and the remainder is appended as the frame-check sequence; the receiver re-divides and a zero remainder indicates no detected errorFig: Hamming (7,4) bit layout: parity P_1,P_2,P_4 at power-of-two positions 1,2,4; data D_1\,-\,D_4 at 3,5,6,7. Each parity bit covers the positions whose binary index includes its weight