Low-density parity-check codes

• Published in: I.R.E. transactions on information theory Vol. 8; no. 1; pp. 21 - 28
• Main Author: Gallager, R.
• Format: Journal Article
• Language: English
• Published: The Institute of Radio Engineers, Inc 01.01.1962
• Subjects: Channel capacity; Communication systems; Data communication; Equations; Error correction codes; Error probability; Information theory; Linear approximation; Maximum likelihood decoding; Parity check codes
• ISSN: 0096-1000
• DOI: 10.1109/TIT.1962.1057683

A low-density parity-check code is a code specified by a parity-check matrix with the following properties: each column contains a small fixed number j \geq 3 of l's and each row contains a small fixed number k > j of l's. The typical minimum distance of these codes increases linearly with block length for a fixed rate and fixed j . When used with maximum likelihood decoding on a sufficiently quiet binary-input symmetric channel, the typical probability of decoding error decreases exponentially with block length for a fixed rate and fixed j . A simple but nonoptimum decoding scheme operating directly from the channel a posteriori probabilities is described. Both the equipment complexity and the data-handling capacity in bits per second of this decoder increase approximately linearly with block length. For j > 3 and a sufficiently low rate, the probability of error using this decoder on a binary symmetric channel is shown to decrease at least exponentially with a root of the block length. Some experimental results show that the actual probability of decoding error is much smaller than this theoretical bound.