Hamming74

Hamming74, or Hamming(7,4), is a linear error-correcting code designed to protect 4 data bits by adding 3 parity bits to form 7-bit codewords. Introduced by Richard Hamming in 1950, it is capable of correcting any single-bit error and detecting most double-bit errors. It has length n = 7, dimension k = 4, and minimum distance d = 3, which yields a coding rate of 4/7.

In the standard arrangement, the seven bit positions are numbered 1 through 7. The three parity bits occupy positions 1, 2, and 4, while the data bits fill positions 3, 5, 6, and 7. The parity bits are chosen so that the parity across certain subsets of bits is even: p1 covers bits 1, 3, 5, 7; p2 covers bits 2, 3, 6, 7; p4 covers bits 4, 5, 6, 7. This structure yields a parity-check matrix H whose columns correspond to the binary representations of the bit positions 1–7.

Decoding uses a syndrome calculation: after receiving a 7-bit word, the three parity checks are evaluated to produce a 3-bit syndrome S, which identifies the position of a single-bit error. If the syndrome is zero, no error occurred. Flipping the bit indicated by the syndrome corrects single-bit errors. If the syndrome is nonzero but the result is not a valid codeword after correction, the error is likely a double-bit error, which the code cannot reliably correct, though such errors are typically detectable.

Hamming74 is a foundational single-error-correcting code in coding theory. It is widely used in educational contexts and serves as a building block for extended Hamming codes that add a parity bit to enable detection of double errors, enhancing reliability in practical systems.

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