Multiple bit errors Hamming(7,4)

a bit error on bit 4 & 5 introduced (shown in blue text) bad parity in green circle (shown in red text)

it not difficult show single bit errors can corrected using scheme. alternatively, hamming codes can used detect single , double bit errors, merely noting product of h nonzero whenever errors have occurred. in adjacent diagram, bits 4 , 5 flipped. yields 1 circle (green) invalid parity errors not recoverable.

however, hamming (7,4) , similar hamming codes cannot distinguish between single-bit errors , two-bit errors. is, two-bit errors appear same one-bit errors. if error correction performed on two-bit error result incorrect.

similarly, hamming codes cannot detect or recover arbitrary three-bit error; consider diagram: if bit in green circle (colored red) 1, parity checking return null vector, indicating there no error in codeword.