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Linear error correcting code
Low-density parity-check (LDPC) codes, also known as Gallager codes, are a class of error-correction codes first proposed in 1960. Together with the closely
Low-density_parity-check_code
Reliable digital data delivery methods on unreliable channels
codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including low-density parity-check codes
Error detection and correction
Error_detection_and_correction
Family of linear error-correcting codes
error has occurred. Coding theory Gray code Golay code Hamming bound Hamming distance Low-density parity-check code Reed–Muller code Reed–Solomon error
Hamming_code
Classification of algorithm
impractical algorithm becomes practical. See, for example, low-density parity-check codes, below. An impractical algorithm can still demonstrate that
Galactic_algorithm
Scheme for controlling errors in data over noisy communication channels
code Linear Network Coding, a type of erasure correcting code across networks instead of point-to-point links Long code Low-density parity-check code
Error_correction_code
A multidimensional parity-check code (MDPC) is a type of error-correcting code that generalizes two-dimensional parity checks to higher dimensions. It
Multidimensional parity-check code
Multidimensional_parity-check_code
Fountain code class
regular low density parity check code. Another possibility would be a concatenation of a Hamming code with a low density parity check code. The inner code takes
Raptor_code
High-performance forward error correction codes
noise. Turbo codes compete with low-density parity-check (LDPC) codes, which provide similar performance. Until the patent for turbo codes expired, the
Turbo_code
Code added to allow recovery of lost data
most popular erasure codes are Reed-Solomon coding, low-density parity-check code (LDPC codes), and turbo codes. As of 2023, modern data storage systems
Erasure_code
System of rules to convert information into another form or representation
Turbo, Golay, algebraic geometry codes, low-density parity-check codes, and space–time codes. Error detecting codes can be optimised to detect burst errors
Code
error-correcting codes are based on sparse graphs, achieving close to the Shannon limit. The archetypal sparse-graph codes are Gallager's low-density parity-check codes
Sparse_graph_code
probabilistic graphical model and thus methods from decoding low-density parity-check codes like belief propagation can be used. In the erasure channel
Sudoku_code
Signals broadcast by GPS satellites
expanded with a 24-bit CRC, then individually encoded using a low-density parity-check code, and then interleaved as a single unit using a block interleaver
GPS_signals
Computer hardware technology that uses quantum mechanics
024-bit or larger. One approach to overcoming errors combines low-density parity-check code with cat qubits that have intrinsic bit-flip error suppression
Quantum_computing
Bipartite graph in coding theory
specify an error correcting code. Tanner graphs play a central role in the design and decoding of low-density parity-check codes. They have also been applied
Tanner_graph
Regius Professor of Engineering at the University of Cambridge (1967–2016)
neural networks, the rediscovery (with Radford M. Neal) of low-density parity-check codes, and the invention of Dasher, a software application for communication
David_J._C._MacKay
Class of error-correcting code
codes, of which BCH codes are an example Reed–Solomon codes Reed–Muller code Algebraic geometry code Binary Goppa code Low-density parity-check codes
Linear_code
Error correction algorithm
modern iteratively-decoded error-correcting codes, including turbo codes and low-density parity-check codes. Based on the trellis: Compute forward probabilities
BCJR_algorithm
Limit on data transfer rate
Advanced techniques such as Reed–Solomon codes and, more recently, low-density parity-check (LDPC) codes and turbo codes, come much closer to reaching the theoretical
Noisy-channel_coding_theorem
Type of error correcting code
compared to well-defined and implemented coding schemes such as low-density parity-check code (LDPC) and turbo code. Polar performance can be improved with
Polar_code_(coding_theory)
Standards for Ethernet at ten times the speed of Gigabit Ethernet
error correction (FEC) coding is performed using a [2048,1723]2 low-density parity-check code on 1723 bits, with the parity check matrix construction based
10_Gigabit_Ethernet
length of the code. In coding theory, an expander code is a [ n , n − m ] 2 {\displaystyle [n,n-m]_{2}\,} linear block code whose parity check matrix is the
Expander_code
Type of two-dimensional barcode
(HCC2D) Code, which builds upon a QR code basis for preserving the QR robustness to distortions and uses colors for increasing data density (as of 2014
QR_code
American electrical engineer (born 1931)
Gallager's 1960 Sc.D. thesis, on low-density parity-check codes, was published by the MIT Press as a monograph in 1963. The codes, which remained useful over
Robert_G._Gallager
Austrian computer scientist
R.L. (Feb 2001). "Design of capacity-approaching irregular low-density parity-check codes". IEEE Transactions on Information Theory. 47 (2): 619–37. doi:10
Rüdiger_Urbanke
Wireless networking standard in the 802.11 family
11n specification: Two to four spatial streams Low-density parity-check code (LDPC) Space–time block coding (STBC) Transmit beamforming (TxBF) 400 ns short
IEEE_802.11ac
Theorem that tells the maximum rate at which information can be transmitted
noisy-channel coding theorem. This text also discusses state-of-the-art methods from coding theory, such as low-density parity-check codes, and Turbo codes. MIT
Shannon–Hartley_theorem
Signal processing technique
of electron beam sensitive materials. Low-density parity-check code Noiselet Sparse approximation Sparse coding Verification-based message-passing algorithms
Compressed_sensing
Home networking over legacy wiring
frequency-division multiplexing (OFDM) modulation and low-density parity-check code (LDPC) forward error correction (FEC) code. G.hn includes the capability to notch
G.hn
Canadian computer scientist and statistician (born 1956)
C.; Neal, R. M. (1996). "Near Shannon limit performance of low density parity check codes". Electronics Letters. 32 (18): 1645. Bibcode:1996ElL....32
Radford_M._Neal
Algorithm for statistical inference on graphical models
empirical success in numerous applications, including low-density parity-check codes, turbo codes, free energy approximation, and satisfiability. The algorithm
Belief_propagation
Electro-mechanical data storage device
newest drives, as of 2009[update], low-density parity-check codes (LDPC) were supplanting Reed–Solomon; LDPC codes enable performance close to the Shannon
Hard_disk_drive
Class of error correction codes
and Computing. Vol. 36. University Of Illinois. pp. 201–210. Iterative Error Correction: Turbo, Low-Density Parity-Check, and Repeat-Accumulate Codes
Repeat-accumulate_code
Power line communications specifications
expensive to implement both HomePlug's turbo coding forward error correction and G.hn's low-density parity-check code. However, IEEE 1901 allows co-existence
HomePlug
Fifth-generation mobile telecommunications standard
5G NR uses polar codes for control channels and low-density parity-check codes (LDPC) for data channels, replacing the turbo codes used in 4G. Research
5G
Self-correcting computer data storage
that would be ignored by a system with no error-checking, would halt a machine with parity checking or be invisibly corrected by ECC: a single bit is
ECC_memory
American computer scientist
Computing Machinery in 2014. His joint paper (Luby et al. 2001) on low-density parity-check codes received the 2002 IEEE Information Theory Society Best Paper
Michael_Mitzenmacher
Mathematical model of memory
Holographic associative memory Low-density parity-check code Memory networks Memory-prediction framework Neural coding Neural Turing machine Random indexing
Sparse_distributed_memory
Next-generation TV standard based on ATSC
orthogonal frequency-division multiplexing (OFDM) modulation with low-density parity-check code (LDPC) forward error correction (FEC). With a 6 MHz channel
ATSC_3.0
Milenkovic is a coding theorist from the former Yugoslavia, known for her work in compressed sensing, low-density parity-check codes, and DNA digital
Olgica_Milenkovic
Electrical Engineer and Computer Scientist (born 1959)
error correction encoders/decoders including turbo codes, low-density parity-check codes, and polar codes. These technologies are found in numerous integrated
Keshab_K._Parhi
American mathematician
in mathematical fields. Her research involves coding theory, including low-density parity-check codes and Tanner graphs. After "a rough start" in a required
Christine_Kelley
American mathematician
Judy L. (2007), "Characterizations of pseudo-codewords of (low-density) parity-check codes", Advances in Mathematics, 213 (1): 205–229, doi:10.1016/j
Judy_L._Walker
Hybrid error-detection and correction code in communications
and Efficiently-Encodable Rate-Compatible (E2RC) code, both of which are low-density parity-check codes. HARQ is used in HSDPA and HSUPA which provide high
Hybrid automatic repeat request
Hybrid_automatic_repeat_request
Chinese lunar communications satellite
itself. With quadrature phase shift keying, encryption with low-density parity check code and a traveling wave tube amplifier with 55 W output power,
Queqiao-2
Convolutional code Viterbi algorithm Soft-decision decoding Interleaver BCJR algorithm Low-density parity-check code Repeat-accumulate code Turbo equalizer
Serial concatenated convolutional codes
Serial_concatenated_convolutional_codes
Quantum key distribution network
near the Shannon limit via forward error correction based on low-density parity-check codes (LDPC). Sifting was performed either by traditional methods
DARPA_Quantum_Network
and Gustave Solomon propose Reed–Solomon codes 1962 – Robert G. Gallager proposes low-density parity-check codes; they are unused for 30 years due to technical
Timeline of information theory
Timeline_of_information_theory
Method for interpreting data in digital storage systems
or BCJR algorithm are essential in iteratively decoding the low-density parity-check code used in modern HDDs. A single integrated circuit contains the
Partial-response maximum-likelihood
Partial-response_maximum-likelihood
surface code quantum error correction, the technique was later extended to other codes such as quantum low density parity check (QLDPC) codes. Current
Lattice_surgery
Energy efficiency enhancement method for digital signal modulation
Steiner and P. Schulte, "Bandwidth Efficient and Rate-Matched Low-Density Parity-Check Coded Modulation," in IEEE Transactions on Communications, vol. 63
Constellation_shaping
a number (0-9); encoding density low: barcode length on 50% longer than Interleaved 2 of 5 symbology and on 172% than Code 128; variable length of symbol;
Industrial_2_of_5
Type of block code
error-correcting codes can be represented as cyclic codes, including BCH codes, Reed-Solomon codes, and some classes of low-density parity-check codes defined
Cyclic_code
System of digitally encoding numbers
BCD code with an extra parity bit applied as least significant bit to achieve odd parity of the resulting 5-bit code is also known as Ferranti code.) Lala
Binary-coded_decimal
French professor in electrical engineering
has been invented for the decoding of low-density parity-check codes (LDPC codes also known as Gallager codes, in honor of Robert G. Gallager, who developed
Claude_Berrou
ARM Cortex-M based Microcontrollers by STMicroelectronics
memories, ECC means memory has error correction code checking, Parity means memory has parity checking. The following Discovery evaluation boards are sold
STM32
Probabilistic model
extraction, speech recognition, computer vision, decoding of low-density parity-check codes, modeling of gene regulatory networks, gene finding and diagnosis
Graphical_model
Problem in information theory and communication
\;1\;1\end{pmatrix}}.} Linear code Syndrome decoding Low-density parity-check code Turbo Code "Distributed source coding for sensor networks" by Z. Xiong
Distributed_source_coding
iteratively-decoded error-correcting codes (in particular low-density parity-check (LDPC) codes and Turbo codes). EXIT charts were developed by Stephan
EXIT_chart
Researchers at the Institute of Science Tokyo discover quantum low-density parity-check codes that asymptotically approach the hashing bound and that can
Timeline of quantum computing and communication
Timeline_of_quantum_computing_and_communication
puzzles Integrity check value Interleaving ISBN ISMN LM hash Lexicographic code Linear code Link adaptation Low-density parity-check Luhn algorithm Luhn
List of algebraic coding theory topics
List_of_algebraic_coding_theory_topics
Encoding methods for representing data on magnetic media
with the contents of the parity track, and the polynomial check code calculated and exclusive-ORed with the received check code, resulting in two 8-bit
Group_coded_recording
American coding theorist
1976) is an American coding theorist of Bosnia and Herzegovina origin. She is known for her work on low-density parity-check codes. She works in the UCLA
Lara_Dolecek
Normalized signal-to-noise ratio measure
discovery of turbo codes, low-density parity-check (LDPC) and polar codes. Chris Heegard and Stephen B. Wicker (1999). Turbo coding. Kluwer. p. 3. ISBN 978-0-7923-8378-9
Eb/N0
algorithms: A GDL like algorithm was used by Gallager's for decoding low density parity-check codes. Based on Gallager's work Tanner introduced the Tanner graph
Generalized_distributive_law
Type of machine learning model
Stereo Set, and Parity Benchmark. Fact-checking and misinformation detection benchmarks are available. A 2023 study compared the fact-checking accuracy of
Large_language_model
fault-tolerant quantum computing design based on quantum low-density parity-check (qLDPC) codes. They show that a 2048-bit RSA integer could be factored
2026_in_science
standard companding algorithm Code-excited linear prediction (CELP): low bit-rate speech compression Linear predictive coding (LPC): lossy compression by
List_of_algorithms
Type of computer memory
memory bits are used to record parity and to enable missing data to be reconstructed by error-correcting code (ECC). Parity allows the detection of all single-bit
Dynamic_random-access_memory
Mitzenmacher. "Verification-based decoding for packet-based low-density parity-check codes." IEEE Transactions on Information Theory 51.1 (2005): 120-127
Verification-based message-passing algorithms in compressed sensing
Verification-based_message-passing_algorithms_in_compressed_sensing
disparity code – PAL – Par meter – Parabolic antenna – Parabolic microphone – Parallel transmission – Parasitic element (electrical networks) – Parity bit –
Index_of_electronics_articles
Method of encoding digital data on multiple carrier frequencies
layer that specifies OFDM with adaptive modulation and a Low-Density Parity-Check (LDPC) FEC code. OFDM is extensively used in wireless LAN and MAN applications
Orthogonal frequency-division multiplexing
Orthogonal_frequency-division_multiplexing
1981 IBM microcomputer
improvement over the 8257 used in the System/23. The RAM memory width and parity check mechanism remained the same, though the memory type changed between projects
IBM_System/23_Datamaster
Digital optical disc format
2011. Cai, Kui (2007). "Introduction" (PDF). Design and Analysis of Parity-Check-Code-Based Optical Recording Systems (Thesis). pp. 1–16. Archived (PDF)
Blu-ray
Class of digital signal-processing methods
employing advanced error-correcting codes that lend themselves to soft decoding, such as low-density parity check (LDPC) codes. For example, if noise-predictive
Noise-predictive maximum-likelihood detection
Noise-predictive_maximum-likelihood_detection
Country primarily in North America
over 25% of nominal global economic output, or 15% at purchasing power parity (PPP). From 1983 to 2008, U.S. real compounded annual GDP growth was 3.3%
United_States
Chinese digital TV standard
noise code (PN) as a guard interval that allows faster synchronization system and a more accurate channel estimation, Low-Density Parity Check (LDPC)
Digital Terrestrial Multimedia Broadcast
Digital_Terrestrial_Multimedia_Broadcast
Type of computer memory introduced 2014
and WE commands, PAR and Alert for error checking and DBI for data bus inversion); New power saving features (low-power auto self-refresh, temperature-controlled
DDR4_SDRAM
Original SCSI storage interface
option of parity bit error checking. Starting with SCSI-U160 (part of SCSI-3) all commands and data are error checked by a cyclic redundancy check. The first
Parallel_SCSI
German mathematical physicist
discovered families of constant-rate and linear-distance low-density parity-check (LDPC) quantum codes correspond to NLTS local Hamiltonians. This result is
Nikolas_Breuckmann
of global economic output. It is the second-largest by purchasing power parity (PPP). On a per capita basis, the U.S. ranks ninth-highest nominal GDP per
Economy_of_the_United_States
IEEE standard for broadband over power lines
Reed–Solomon (RS) and convolutional code, and an option to use low-density parity-check (LDPC) code. On top of these two physical layers, two different media
IEEE_1901
System/360) IBM 3400-4: Lower density tape IBM 3400-6: Normal tape IBM 3410: Magnetic tape drive (9 Track—8 data bits & 1 parity bit); 1971 IBM 3411: Magnetic
List_of_IBM_products
Country in Central Europe
economies". As of 2023[update], the country's GDP per capita at purchasing power parity is $51,329 and $29,856 at nominal value. According to Allianz A.G., in 2024
Czech_Republic
Type of computing error
correcting code. Alternatively, roll-back error correction can be used, detecting the soft error with an error-detecting code such as parity, and rewriting
Soft_error
Country in Central Europe
international pressure to reduce banking secrecy and raise tax rates to parity with the EU. Preparatory discussions involved four areas: the electricity
Switzerland
Highest rate quantum information can be sent through a noisy quantum channel
constructions. In 2025, Daiki Komoto and Kenta Kasai reported low-density parity-check CSS codes for the depolarizing channel whose numerical decoding performance
Quantum_capacity
Intel microprocessor
Developers Manual, Vol 1). Enhanced self-test features like the L1 cache parity check (see Cache Structure in the Developers Manual, Vol 1). Other built-in
Pentium_(original)
Dimensions and location of feed holes and code holes ISO 1155:1978 Information processing – Use of longitudinal parity to detect errors in information messages
List_of_ISO_standards_1–1999
Country in South America
history, Paraguay has been a recipient of immigrants owing to its low population density, especially after the demographic collapse caused by the Paraguayan
Paraguay
Country in South America
"CIA – The World Factbook – Country Comparisons – GDP (purchasing power parity)". Central Intelligence Agency. Archived from the original on 4 June 2011
Brazil
Country in South Asia
nationwide. Pakistan's economy ranks 27th globally by purchasing power parity (PPP) and 42nd by nominal GDP. Historically, Pakistan was part of the wealthiest
Pakistan
List of Android operating system versions
operating system itself and the SDK were released along with their source code, as free software under the Apache License. The first public release of Android
Android_version_history
Flat, usually circular disc that encodes binary data
measurements, including so-called "C1", "C2" and "CU" errors on CDs, and "PI/PO (parity inner/outer) errors" and the more critical "PI/PO failures" on DVDs. Finer-grain
Optical_disc
Second revision of the DVB-T standard
or 192nd grade Inner encoding: a Low Density Parity Check (LDPC) code is cascaded to the BCH Bit interleaving Parity bits block interleaving Twist column
DVB-T2
IBM mainframe operating system designed for use with smaller machines
unassigned. <tape option> specifies either tape mode settings such as density, parity, etc., or "ALT" to indicate an alternate device. The RESET statement
DOS/360_and_successors
Country in Central and South Asia
nominal GDP was $20.1 billion in 2020, or $81 billion by purchasing power parity (PPP). Its GDP per capita is $2,459 (PPP) and $611 by nominal. Despite having
Afghanistan
Country in Southern Africa
compared to other countries in sub-Saharan Africa, and its purchasing power parity (PPP) of around US$16,000 ranks among the highest on the continent as of
South_Africa
Experimental technology level
simulate. IBM's approach relies on quantum low-density parity check (qLDPC) codes and bivariate bicycle codes to minimize the physical qubit overhead required
Noisy intermediate-scale quantum computing
Noisy_intermediate-scale_quantum_computing
Small motorcycle-like motor road vehicle
and particulate emissions of the Euro 3 automobile standards. Approximate parity with automobiles was achieved with NOx emissions in these studies. Emissions
Moped
LOW DENSITY-PARITY-CHECK-CODE
LOW DENSITY-PARITY-CHECK-CODE
Girl/Female
Hindu
Like Paris, Fairy or like a fairy
Male
English
English short form of Latin Hector, HECK means "defend; hold fast."
Male
English
Middle English short form of English Lawrence, LAW means "of Laurentum."
Male
Italian
Italian form of Greek Paris, probably PARIDE means "wager."Â
Girl/Female
American, Australian, British, Christian, English, German, Indian, Latin
Act of Kindness; Charity; Brotherly Love; Affection
Female
English
English name derived from the vocabulary word clarity, from Latin clarus, CLARITY means "clear."
Male
Greek
(Λώτ) Greek form of Hebrew Lowt, LOT means "covering, veil." In the bible, this is the name of a nephew of Abraham and father of Moab.
Girl/Female
British, English
Purity
Surname or Lastname
English and Scottish
English and Scottish : topographic name for someone who lived near a tumulus, mound or hill, Middle English lowe, from Old English hlÄw (see Law 2).Scottish and English : nickname for a short man, from Middle English lah, lowe (Old Norse lágr; the word was adopted first into the northern dialects of Middle English, where Scandinavian influence was strong, and then spread south, with regular alteration of the vowel quality).English and Scottish (of Norman origin) : nickname for a violent or dangerous person, from Anglo-Norman French lou, leu ‘wolf’ (Latin lupus). Wolves were relatively common in Britain at the time when most surnames were formed, as there still existed large tracts of uncleared forest.Scottish : from a pet form of Lawrence. Compare Lowry 1.Americanized spelling of Jewish Lowe.
Boy/Male
English American
A from the Old English 'ceorl' meaning man. Famous bearer: American singer Chuck Berry.
Boy/Male
Tamil
Check, Control
Surname or Lastname
English
English : variant spelling of Cheek.
Male
English
Pet form of English Charles, CHUCK means "man."
Boy/Male
British, English
Laurel
Male
English
 English short form of Spanish Alonso, LON means "noble and ready." Compare with another form of Lon.
Boy/Male
Hindu
Check, Control
Surname or Lastname
Possibly an Americanized spelling of Czech and Slovak ÄŒech (see Cech), or other Slavic or German ethnic names for a Czech.English
Possibly an Americanized spelling of Czech and Slovak ÄŒech (see Cech), or other Slavic or German ethnic names for a Czech.English : unexplained.
Male
Polish
 Polish form of Yiddish Lev, LEW means "lion." Compare with another form of Lew.
Surname or Lastname
English
English : from Middle English chike ‘young fowl’ (a shortened form of chiken), applied as a metonymic occupational name for someone who bred poultry for the table, or as a nickname from the same word used as a term of endearment.English : variant of Cheek.
Male
English
 Short form of English Lewis, LEW means "famous warrior." Compare with another form of Lew.
LOW DENSITY-PARITY-CHECK-CODE
LOW DENSITY-PARITY-CHECK-CODE
Girl/Female
Hindu, Indian
Sight
Boy/Male
Arabic, Muslim
Servant of the Exalted (Allah)
Girl/Female
Hindu
Modesty, Education
Boy/Male
Hindu, Indian
Green
Boy/Male
Arabic, Muslim
Servant of the Trustee
Boy/Male
Arabic, Hindu, Indian, Malaysian, Muslim, Punjabi, Sikh, Sindhi
Divine Light; Attribute of God
Male
Spanish
Spanish and Portuguese form of Latin Eligius, ELOY means "to choose."
Girl/Female
Latin
Daughter of Chryses.
Boy/Male
Muslim/Islamic
Unique
Boy/Male
English
free men's town.
LOW DENSITY-PARITY-CHECK-CODE
LOW DENSITY-PARITY-CHECK-CODE
LOW DENSITY-PARITY-CHECK-CODE
LOW DENSITY-PARITY-CHECK-CODE
LOW DENSITY-PARITY-CHECK-CODE
adv.
With a low voice or sound; not loudly; gently; as, to speak low.
v. t.
To make checks or chinks in; to cause to crack; as, the sun checks timber.
n.
The principles of the low-church party.
n.
A wedge, or brake, to check the motion of a machine or car; a chock.
superl.
Depressed in the scale of sounds; grave; as, a low pitch; a low note.
n.
The cheek bone.
superl.
Deficient in vital energy; feeble; weak; as, a low pulse; made low by sickness.
v. t.
To verify, to guard, to make secure, by means of a mark, token, or other check; to distinguish by a check; to put a mark against (an item) after comparing with an original or a counterpart in order to secure accuracy; as, to check an account; to check baggage.
superl.
Not loud; as, a low voice; a low sound.
n.
The quality or state of being rare; rareness; thinness; as, the rarity (contrasted with the density) of gases.
v. t.
To make a move which puts an adversary's piece, esp. his king, in check; to put in check.
n.
A low-lying piece of watery land; -- called also flow moss and flow bog.
n.
Small chick or crack.
adv.
Entirely; quite; as, chock home; chock aft.
superl.
Moderate; not intense; not inflammatory; as, low heat; a low temperature; a low fever.
n.
Rarily; rareness; thinness, as of a fluid; as, the tenuity of the air; the tenuity of the blood.
a.
Checkered; designed in checks.
v. t.
A pattern in checks; a single check.
superl.
Wanting strength or animation; depressed; dejected; as, low spirits; low in spirits.
n.
A mark, certificate, or token, by which, errors may be prevented, or a thing or person may be identified; as, checks placed against items in an account; a check given for baggage; a return check on a railroad.