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Hamming(7,4) Error-Correcting Code

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Overview: Hamming(7,4) encodes 4 bits into 7 bits with 3 parity bits
Richard W. Hamming introduced it in 1950 at Bell Labs
Can correct single-bit errors or detect all single-bit and two-bit errors
Minimal Hamming distance between correct codewords is 3
Mathematical Structure: Uses code generator matrix G and parity-check matrix H
Data bits are mapped to parity bits in rows 1-4 of G
Remaining rows form identity matrix
Syndrome vector H indicates error detection at receiver
Error Detection: Error-free transmission results in null syndrome vector
Single-bit errors detected by checking parity of affected circles
Two-bit errors appear identical to single-bit errors
Cannot detect or correct arbitrary three-bit errors
Applications: Used as base for Steane code in quantum information
Closely related to E7 lattice
Can be extended to 8-bit code with additional parity bit