{"doi":"10.1109/tit.1962.1057683","title":"Low-density parity-check codes","abstract":"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 <tex xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">j \\geq 3</tex> of l's and each row contains a small fixed number <tex xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">k &gt; j</tex> of l's. The typical minimum distance of these codes increases linearly with block length for a fixed rate and fixed <tex xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">j</tex> . 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 <tex xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">j</tex> . 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 <tex xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">j &gt; 3</tex> 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.","journal":"IEEE Transactions on Information Theory","year":1962,"id":12062,"datarank":36.453785671522155,"base_score":9.266342562091468,"endowment":9.266342562091468,"self_citation_contribution":1.3899513843137206,"citation_network_contribution":35.06383428720844,"self_endowment_contribution":1.3899513843137206,"citer_contribution":35.06383428720844,"corpus_percentile":99.4,"corpus_rank":2096,"citation_count":10575,"citer_count":199,"citers_with_citation_signal":199,"citers_with_endowment":199,"datacite_reuse_total":0,"is_dataset":false,"is_oa":false,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"1962-01-01","authors":[{"id":96101,"name":"Robert G. Gallager","orcid":"0000-0003-4445-2917","position":1,"is_corresponding":false},{"id":96100,"name":"R. Gallager","orcid":null,"position":0,"is_corresponding":true}],"reference_count":8,"raw_metadata":{"citation_network_status":"fetched"},"created_at":"2026-03-01T18:20:47.508186Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}