391:
403:
564:
whose error rate tends to zero: His proof relies on
Gaussian random coding, which is not suitable to real-world applications. The upper bound given by Shannon's work inspired a long journey in designing ECCs that can come close to the ultimate performance boundary. Various codes today can attain almost the Shannon limit. However, capacity achieving ECCs are usually extremely complex to implement.
808:
560:
the receiver SNR (signal-to-noise-ratio) decreasing the bit error rate, at the cost of reducing the effective data rate. On the other extreme, not using any ECC (i.e., a code-rate equal to 1) uses the full channel for information transfer purposes, at the cost of leaving the bits without any additional protection.
953:
Transmitted sentence: ThisIsAnExampleOfInterleaving... Error-free transmission: TIEpfeaghsxlIrv.iAaenli.snmOten. Received sentence with a burst error: TIEpfe______Irv.iAaenli.snmOten. Received sentence after deinterleaving:
586:
coding schemes in which a short constraint-length
Viterbi-decoded convolutional code does most of the work and a block code (usually Reed–Solomon) with larger symbol size and block length "mops up" any errors made by the convolutional decoder. Single pass decoding with this family of error correction
441:
increasing complexity. A convolutional code that is terminated is also a 'block code' in that it encodes a block of input data, but the block size of a convolutional code is generally arbitrary, while block codes have a fixed size dictated by their algebraic characteristics. Types of termination for
762:
Sometimes it is only necessary to decode single bits of the message, or to check whether a given signal is a codeword, and do so without looking at the entire signal. This can make sense in a streaming setting, where codewords are too large to be classically decoded fast enough and where only a few
563:
One interesting question is the following: how efficient in terms of information transfer can an ECC be that has a negligible decoding error rate? This question was answered by Claude
Shannon with his second theorem, which says that the channel capacity is the maximum bit rate achievable by any ECC
559:
The redundant bits that protect the information have to be transferred using the same communication resources that they are trying to protect. This causes a fundamental tradeoff between reliability and data rate. In one extreme, a strong code (with low code-rate) can induce an important increase in
555:
The fundamental principle of ECC is to add redundant bits in order to help the decoder to find out the true message that was encoded by the transmitter. The code-rate of a given ECC system is defined as the ratio between the number of information bits and the total number of bits (i.e., information
182:
can be used to compute the maximum achievable communication bandwidth for a given maximum acceptable error probability. This establishes bounds on the theoretical maximum information transfer rate of a channel with some given base noise level. However, the proof is not constructive, and hence gives
965:
Use of interleaving techniques increases total delay. This is because the entire interleaved block must be received before the packets can be decoded. Also interleavers hide the structure of errors; without an interleaver, more advanced decoding algorithms can take advantage of the error structure
987:
context. The idea is to directly use software ECCs in the communications. For instance in the 5G, the software ECCs could be located in the cloud and the antennas connected to this computing resources: improving this way the flexibility of the communication network and eventually increasing the
567:
The most popular ECCs have a trade-off between performance and computational complexity. Usually, their parameters give a range of possible code rates, which can be optimized depending on the scenario. Usually, this optimization is done in order to achieve a low decoding error probability while
928:
Error-free code words: aaaabbbbccccddddeeeeffffgggg
Interleaved: abcdefgabcdefgabcdefgabcdefg Transmission with a burst error: abcdefgabcd____bcdefgabcdefg Received code words after deinterleaving:
169:
The maximum proportion of errors or missing bits that can be corrected is determined by the design of the ECC, so different forward error correcting codes are suitable for different conditions. In general, a stronger code induces more redundancy that needs to be transmitted using the available
320:
ECC could be said to work by "averaging noise"; since each data bit affects many transmitted symbols, the corruption of some symbols by noise usually allows the original user data to be extracted from the other, uncorrupted received symbols that also depend on the same user data.
199:
to the transmitted information using an algorithm. A redundant bit may be a complicated function of many original information bits. The original information may or may not appear literally in the encoded output; codes that include the unmodified input in the output are
556:
plus redundancy bits) in a given communication package. The code-rate is hence a real number. A low code-rate close to zero implies a strong code that uses many redundant bits to achieve a good performance, while a large code-rate close to 1 implies a weak code.
777:
are error-correcting codes for which single bits of the message can be probabilistically recovered by only looking at a small (say constant) number of positions of a codeword, even after the codeword has been corrupted at some constant fraction of positions.
914:
Here, each group of the same letter represents a 4-bit one-bit error-correcting codeword. The codeword cccc is altered in one bit and can be corrected, but the codeword dddd is altered in three bits, so either it cannot be decoded at all or it might be
998:(A Fast Forward Error Correction Toolbox): a full communication chain in C++ (many supported codes like Turbo, LDPC, Polar codes, etc.), very fast and specialized on channel coding (can be used as a program for simulations or as a library for the SDR).
1669:
Both Reed–Solomon algorithm and BCH algorithm are common ECC choices for MLC NAND flash. ... Hamming based block codes are the most commonly used ECC for SLC.... both Reed–Solomon and BCH are able to handle multiple errors and are widely used on MLC
311:
is a relatively inefficient ECC. Better ECC codes typically examine the last several tens or even the last several hundreds of previously received bits to determine how to decode the current small handful of bits (typically in groups of 2 to 8 bits).
978:
Simulating the behaviour of error-correcting codes (ECCs) in software is a common practice to design, validate and improve ECCs. The upcoming wireless 5G standard raises a new range of applications for the software ECCs: the
623:(the theoretical maximum) using an iterated soft-decision decoding approach, at linear time complexity in terms of their block length. Practical implementations rely heavily on decoding the constituent SPC codes in parallel.
2168:"Digital Video Broadcast (DVB); Second generation framing structure, channel coding and modulation systems for Broadcasting, Interactive Services, News Gathering and other satellite broadband applications (DVB-S2)".
827:
exceeds the error-correcting code's capability, it fails to recover the original code word. Interleaving alleviates this problem by shuffling source symbols across several code words, thereby creating a more
347:
Interleaving ECC coded data can reduce the all or nothing properties of transmitted ECC codes when the channel errors tend to occur in bursts. However, this method has limits; it is best used on narrowband
689:
is an iterated soft-decoding scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the
437:, though other algorithms are sometimes used. Viterbi decoding allows asymptotically optimal decoding efficiency with increasing constraint length of the convolutional code, but at the expense of
850:
For turbo codes, an interleaver is an integral component and its proper design is crucial for good performance. The iterative decoding algorithm works best when there are not short cycles in the
500:
algorithms, which means that for every input and output signal a hard decision is made whether it corresponds to a one or a zero bit. In contrast, convolutional codes are typically decoded using
568:
minimizing the impact to the data rate. Another criterion for optimizing the code rate is to balance low error rate and retransmissions number in order to the energy cost of the communication.
1641:
125:
Long-latency connections also benefit; in the case of satellites orbiting distant planets, retransmission due to errors would create a delay of several hours. FEC is also widely used in
1709:
136:
FEC processing in a receiver may be applied to a digital bit stream or in the demodulation of a digitally modulated carrier. For the latter, FEC is an integral part of the initial
870:
S-random interleaver (where the interleaver is a known random permutation with the constraint that no input symbols within distance S appear within a distance of S in the output).
782:
are error-correcting codes for which it can be checked probabilistically whether a signal is close to a codeword by only looking at a small number of positions of the signal.
847:, typically assumes an independent distribution of errors. Systems using LDPC codes therefore typically employ additional interleaving across the symbols within a code word.
788:
Not all locally decodable codes (LDCs) are locally testable codes (LTCs) neither locally correctable codes (LCCs), q-query LCCs are bounded exponentially while LDCs can have
2187:
Andrews, K. S.; Divsalar, D.; Dolinar, S.; Hamkins, J.; Jones, C. R.; Pollara, F. (November 2007). "The
Development of Turbo and LDPC Codes for Deep-Space Applications".
372:
uses a variety of ECC rates, adding more error-correction bits per packet when there are higher error rates in the channel, or taking them out when they are not needed.
1361:
2328:"Digital Video Broadcast (DVB); Frame structure, channel coding and modulation for a second generation digital terrestrial television broadcasting system (DVB-T2)".
941:
Original transmitted sentence: ThisIsAnExampleOfInterleaving
Received sentence with a burst error: ThisIs______pleOfInterleaving
118:
FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in
911:
Error-free message: aaaabbbbccccddddeeeeffffgggg
Transmission with a burst error: aaaabbbbccc____deeeeffffgggg
1381:
2674:
661:(ITU-T Standard for networking over power lines, phone lines and coaxial cable). Other LDPC codes are standardized for wireless communication standards within
619:(LDPC) codes are a class of highly efficient linear block codes made from many single parity check (SPC) codes. They can provide performance very close to the
932:
In each of the codewords "aaaa", "eeee", "ffff", and "gggg", only one bit is altered, so one-bit error-correcting code will decode everything correctly.
360:, and then fail to work at all if the bit error rate is ever worse. However, some systems adapt to the given channel error conditions: some instances of
296:
This allows an error in any one of the three samples to be corrected by "majority vote", or "democratic voting". The correcting ability of this ECC is:
1648:. 2005. Both say: "The Hamming algorithm is an industry-accepted method for error detection and correction in many SLC NAND flash-based applications."
1487:
2737:
2263:
1638:
1384:
508:
algorithms, which process (discretized) analog signals, and which allow for much higher error-correction performance than hard-decision decoding.
426:
Block codes work on fixed-size blocks (packets) of bits or symbols of predetermined size. Practical block codes can generally be hard-decoded in
815:
Interleaving is frequently used in digital communication and storage systems to improve the performance of forward error correcting codes. Many
537:. A few forward error correction codes are designed to correct bit-insertions and bit-deletions, such as Marker Codes and Watermark Codes. The
2596:
2545:
2134:
2074:
1982:
1850:
2532:. North-Holland Mathematical Library. Vol. 16 (digital print of 12th impression, 1st ed.). Amsterdam / London / New York / Tokyo:
1869:
1452:
1416:
2687:
2680:
1630:
2668:
1905:
966:
and achieve more reliable communication than a simpler decoder combined with an interleaver. An example of such an algorithm is based on
577:
518:
In contrast to classical block codes that often specify an error-detecting or error-correcting ability, many modern block codes such as
1004:: a C++ library of classes and functions for linear algebra, numerical optimization, signal processing, communications, and statistics.
587:
codes can yield very low error rates, but for long range transmission conditions (like deep space) iterative decoding is recommended.
485:
memory errors. This provides single-bit error correction and 2-bit error detection. Hamming codes are only suitable for more reliable
471:
1271:
447:
2659:
2636:
2604:
2577:
2501:
1555:
829:
768:
187:
come very close to the theoretical maximum given by the
Shannon channel capacity under the hypothesis of an infinite length frame.
2533:
602:
craft used iterative concatenated codes to compensate for the very high error rate conditions caused by having a failed antenna.
493:(MLC) NAND may use multi-bit correcting ECC such as BCH or Reed–Solomon. NOR Flash typically does not use any error correction.
1739:
764:
361:
196:
81:
1411:
369:
2224:
Dolinar, S.; Divsalar, D. (15 August 1995). "Weight
Distributions for Turbo Codes Using Random and Nonrandom Permutations".
325:
Because of this "risk-pooling" effect, digital communication systems that use ECC tend to work well above a certain minimum
630:
in his PhD thesis in 1960, but due to the computational effort in implementing encoder and decoder and the introduction of
1792:
1579:
1215:
616:
611:
2521:
1661:
175:
1930:
Kothari, Pravesh K.; Manohar, Peter (2023). "An
Exponential Lower Bound for Linear 3-Query Locally Correctable Codes".
2525:
1396:
1159:
833:
463:
137:
1769:
1257:
894:
184:
1290:
1041:
824:
433:
Convolutional codes work on bit or symbol streams of arbitrary length. They are most often soft decoded with the
308:
1784:
1426:
878:
717:
526:
365:
1316:
183:
no insight of how to build a capacity achieving code. After years of research, some advanced FEC systems like
159:(magnetic, optical and solid state/flash based) devices to enable recovery of corrupted data, and is used as
2189:
1321:
1242:
1210:
1117:
1081:
148:
to demodulate digital data from an analog signal corrupted by noise. Many FEC decoders can also generate a
2241:
1574:
1281:
984:
874:
774:
753:
2153:; Mitzenmacher, M.; Shokrollahi, A.; Spielman, D.; Stemann, V. (1997). "Practical Loss-Resilient Codes".
1499:
716:, and other carriers. It is also used for the evolution of CDMA2000 1x specifically for Internet access,
1431:
1204:
1173:
816:
779:
757:
402:
326:
171:
145:
74:
1276:
1683:
2396:
2282:
2233:
1139:
599:
538:
104:
to request re-transmission may not be needed. The cost is a fixed, higher forward channel bandwidth.
100:
that may occur anywhere in the message, but often to correct a limited number of errors. Therefore a
2261:
Takeshita, Oscar (2006). "Permutation Polynomial Interleavers: An Algebraic-Geometric Perspective".
2246:
2595:
Series in Computer Systems. Vol. 10 (1 ed.). Cambridge, Massachusetts, USA / London, UK:
801:
789:
31:
2386:
2298:
2272:
2206:
2092:
2080:
2035:
1988:
1960:
1931:
1866:
1705:"A Simple Scheme for Belief Propagation Decoding of BCH and RS Codes in Multimedia Transmissions"
1610:
1199:
1144:
991:
In this context, there are various available Open-source software listed below (non exhaustive).
627:
419:
407:
385:
112:
46:
1464:
861:
rectangular (or uniform) interleavers (similar to the method using skip factors described above)
590:
Concatenated codes have been standard practice in satellite and deep space communications since
340:– becomes more pronounced as stronger codes are used that more closely approach the theoretical
1627:
1327:
1010:: implementation (in C) of the 3GPP specifications concerning the Evolved Packet Core Networks.
2655:
2632:
2610:
2600:
2573:
2551:
2541:
2497:
2424:
2155:
Proc. 29th Annual Association for Computing Machinery (ACM) Symposium on Theory of Computation
2130:
2070:
2027:
1978:
1890:
1846:
1551:
1223:
1169:
1163:
1103:
1092:
763:
bits of the message are of interest for now. Also such codes have become an important tool in
486:
434:
390:
70:
42:
631:
522:
lack such guarantees. Instead, modern codes are evaluated in terms of their bit error rates.
2487:
2479:
2451:
2414:
2404:
2290:
2198:
2062:
2019:
1970:
1838:
1809:
1801:
1718:
1600:
1592:
1183:
957:
No word is completely lost and the missing letters can be recovered with minimal guesswork.
890:
725:
709:
530:
529:
codes correct only bit-flips, but not bit-insertions or bit-deletions. In this setting, the
490:
438:
213:
A simplistic example of ECC is to transmit each data bit 3 times, which is known as a (3,1)
130:
111:
pioneered this field in the 1940s and invented the first error-correcting code in 1950: the
217:. Through a noisy channel, a receiver might see 8 versions of the output, see table below.
2126:
2104:
1873:
1814:
1645:
1634:
1286:
591:
459:
427:
214:
202:
163:
152:(BER) signal which can be used as feedback to fine-tune the analog receiving electronics.
141:
108:
101:
97:
1691:
For SLC, a code with a correction threshold of 1 is sufficient. t=4 required ... for MLC.
889:
communication systems, interleaving across carriers may be employed to provide frequency
2400:
2286:
2237:
2419:
2374:
1805:
1596:
1366:
967:
886:
534:
505:
357:
179:
170:
bandwidth, which reduces the effective bit-rate while improving the received effective
149:
2456:
1528:
2731:
2651:
2628:
2084:
1306:
1266:
1252:
1232:
1219:
1178:
1149:
729:
713:
690:
669:
620:
582:
Classical (algebraic) block codes and convolutional codes are frequently combined in
341:
66:
50:
2686:
Sphere Packings, Lattices and Groups, By J. H. Conway, Neil James Alexander Sloane,
2348:
2315:
2039:
1992:
1953:"Exponential lower bound for 2-query locally decodable codes via a quantum argument"
1614:
364:
use a fixed ECC method as long as the ECC can handle the error rate, then switch to
2569:
2210:
2150:
1886:
1547:
1406:
1302:
1298:
1237:
1207:, a type of erasure correcting code across networks instead of point-to-point links
1193:
1188:
1052:
851:
650:
646:
637:
LDPC codes are now used in many recent high-speed communication standards, such as
512:
475:
451:
395:
353:
337:
156:
1131:
used for radar, telemetry, ultra sound, Wifi, DSSS mobile phone networks, GPS etc.
2054:
2007:
1831:"Optimizing the code rate for achieving energy-efficient wireless communications"
1125:, which can be designed to correct any arbitrary number of errors per code block.
2537:
1835:
Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC)
1421:
1262:
1248:
1134:
1128:
820:
2409:
2302:
1952:
1830:
541:
is a more appropriate way to measure the bit error rate when using such codes.
410:
where redundant bits are added continuously into the structure of the code word
2202:
1842:
1311:
1154:
854:
that represents the decoder; the interleaver is chosen to avoid short cycles.
840:
694:
686:
681:
519:
482:
415:
381:
160:
2471:
2031:
1639:"Micron Technical Note TN-29-08: Hamming Codes for NAND Flash Memory Devices"
1295:
Spinal code, a rateless, nonlinear code based on pseudo-random hash functions
944:
The term "AnExample" ends up mostly unintelligible and difficult to correct.
398:) where redundant bits are added as a block to the end of the initial message
2592:
2483:
2294:
2066:
2023:
1401:
844:
119:
38:
2528:(2007) . Written at AT&T Shannon Labs, Florham Park, New Jersey, USA.
2428:
1957:
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
1007:
807:
17:
2059:
Proceedings of the forty-first annual ACM symposium on Theory of computing
1974:
1723:
1704:
1965:
1122:
1112:
867:
random interleavers (where the interleaver is a known random permutation)
721:
705:
701:
697:
in terms of practical application, they now provide similar performance.
550:
467:
2492:
1196:
for non-white noise (prevalent for example in broadband over powerlines)
515:. Hence classical block codes are often referred to as algebraic codes.
1605:
1588:
1228:
916:
654:
27:
Scheme for controlling errors in data over noisy communication channels
2722:
2681:
Observations on Errors, Corrections, & Trust of Dependent Systems
2442:
Nordstrom, A.W.; Robinson, J.P. (1967), "An optimum nonlinear code",
1060:
1001:
638:
595:
700:
One of the earliest commercial applications of turbo coding was the
2391:
2277:
1936:
511:
Nearly all classical block codes apply the algebraic properties of
2716:
2008:"Towards 3-query locally decodable codes of subexponential length"
980:
806:
642:
401:
389:
126:
1906:"'Magical' Error Correction Scheme Proved Inherently Inefficient"
736:, Sprint's consumer and business marketing names for 1xEV-DO are
649:
standard for microwave communications), High-Speed Wireless LAN (
2669:"Error Correction Code in Single Level Cell NAND Flash memories"
2614:
2555:
2375:"Deep Neural Network Probabilistic Decoder for Stabilizer Codes"
2333:
2173:
785:
Not all testing codes are locally decoding and testing of codes
665:
662:
658:
2589:
A Commonsense Approach to the Theory of Error Correcting Codes
2476:
Proceedings of the 10th ACM Workshop on Hot Topics in Networks
800:"Interleaver" redirects here. For the fiber-optic device, see
732:, and other carriers (Verizon's marketing name for 1xEV-DO is
641:(Digital Video Broadcasting – Satellite – Second Generation),
455:
30:"Interleaver" redirects here. For the fiber-optic device, see
442:
convolutional codes include "tail-biting" and "bit-flushing".
80:
The central idea is that the sender encodes the message in a
2470:
Perry, Jonathan; Balakrishnan, Hari; Shah, Devavrat (2011).
1577:(April 1950). "Error Detecting and Error Correcting Codes".
1486:
Charles Wang; Dean Sklar; Diana Johnson (Winter 2001–2002).
1451:
Charles Wang; Dean Sklar; Diana Johnson (Winter 2001–2002).
995:
823:
rather than independently. If the number of errors within a
545:
Code-rate and the tradeoff between reliability and data rate
166:
on systems that require special provisions for reliability.
2625:
Error Control Systems for Digital Communication and Storage
303:
up to 2 bits of triplet omitted (cases not shown in table).
2723:
lpdec: library for LP decoding and related things (Python)
2702:
2055:"3-query locally decodable codes of subexponential length"
1385:
Mathematics of cyclic redundancy checks § Bitfilters
1829:
Rosas, F.; Brante, G.; Souza, R. D.; Oberli, C. (2014).
1710:
International Journal of Digital Multimedia Broadcasting
704:(TIA IS-2000) digital cellular technology developed by
1363:
bits long for optimal generator polynomials of degree
832:
of errors. Therefore, interleaving is widely used for
1369:
1330:
1324:(CRCs) can correct 1-bit errors for messages at most
2373:
Krastanov, Stefan; Jiang, Liang (8 September 2017).
1951:
Kerenidis, Iordanis; de Wolf, Ronald (9 June 2003).
594:
first used the technique in its 1986 encounter with
2116:
2114:
1662:"What Types of ECC Should Be Used on Flash Memory?"
2566:Error-Correction Coding for Digital Communications
1738:Shah, Gaurav; Molina, Andres; Blaze, Matt (2006).
1375:
1355:
462:. Other examples of classical block codes include
96:). The redundancy allows the receiver not only to
634:codes, they were mostly ignored until the 1990s.
307:Though simple to implement and widely used, this
1876:, IEEE, 29 October 2009, accessed 21 March 2011.
1233:rateless erasure correcting code (Fountain code)
496:Classical block codes are usually decoded using
1544:Practical Error Correction Design For Engineers
572:Concatenated ECC codes for improved performance
2675:"Error Correction Code in NAND Flash memories"
1891:"Locally Testable vs. Locally Decodable Codes"
1684:"The Inconvenient Truths of NAND Flash Memory"
1498:(1). The Aerospace Corporation. Archived from
1463:(1). The Aerospace Corporation. Archived from
819:are not memoryless: errors typically occur in
720:(TIA IS-856). Like 1x, EV-DO was developed by
2587:Arazi, Benjamin (1987). Swetman, Herb (ed.).
2564:Clark, Jr., George C.; Cain, J. Bibb (1981).
1628:"Hamming codes for NAND flash memory devices"
1078:6 (double-error correct-/triple error detect)
356:designed to tolerate the expected worst-case
8:
839:The analysis of modern iterated codes, like
352:Most telecommunication systems use a fixed
1656:
1654:
2491:
2455:
2418:
2408:
2390:
2276:
2245:
1964:
1935:
1813:
1722:
1604:
1569:
1567:
1368:
1335:
1329:
811:A short illustration of interleaving idea
414:The two main categories of ECC codes are
2123:Turbo codes: principles and applications
1785:"A mathematical theory of communication"
1512:How Forward Error-Correcting Codes Work]
1018:
481:Hamming ECC is commonly used to correct
450:is noteworthy for its widespread use in
219:
2690:, 2013-03-09 – Mathematics – 682 pages.
2264:IEEE Transactions on Information Theory
1546:(Revision 1.1, 2nd ed.). CO, USA:
1443:
2100:
2090:
1766:Fundamentals of Wireless Communication
1764:Tse, David; Viswanath, Pramod (2005),
954:T_isI_AnE_amp_eOfInterle_vin_...
533:is the appropriate way to measure the
2719:. Database of error correcting codes.
2688:Springer Science & Business Media
2650:. Englewood Cliffs, New Jersey, USA:
2627:. Englewood Cliffs, New Jersey, USA:
2597:Massachusetts Institute of Technology
1637:. EE Times-Asia. Apparently based on
446:There are many types of block codes;
7:
2530:The Theory of Error-Correcting Codes
1542:Glover, Neal; Dudley, Trent (1990).
1522:
1520:
1417:Error-correcting codes with feedback
626:LDPC codes were first introduced by
504:algorithms like the Viterbi, MAP or
1245:, used in Geometry and Group Theory
981:Cloud Radio Access Networks (C-RAN)
974:Software for error-correcting codes
877:(QPP). An example of use is in the
748:Local decoding and testing of codes
578:Concatenated error correction codes
368:when the error rate gets too high;
300:Up to 1 bit of triplet in error, or
2006:Yekhanin, Sergey (February 2008).
1806:10.1002/j.1538-7305.1948.tb01338.x
1703:Baldi, M.; Chiaraluce, F. (2008).
1597:10.1002/j.1538-7305.1950.tb00463.x
881:mobile telecommunication standard.
769:probabilistically checkable proofs
25:
2703:"The Correcting Codes (ECC) Page"
2701:Morelos-Zaragoza, Robert (2004).
1865:IEEE Standard, section 20.3.11.6
1488:"Forward Error-Correction Coding"
1453:"Forward Error-Correction Coding"
988:energy efficiency of the system.
936:Transmission without interleaving
906:Transmission without interleaving
1904:Brubaker, Ben (9 January 2024).
1267:rateless erasure correcting code
1253:rateless erasure correcting code
551:Bit rate § Information rate
316:Averaging noise to reduce errors
2683:, by James Hamilton, 2012-02-26
2053:Efremenko, Klim (31 May 2009).
1740:"Keyboards and covert channels"
765:computational complexity theory
606:Low-density parity-check (LDPC)
362:hybrid automatic repeat-request
2738:Error detection and correction
2121:Vucetic, B.; Yuan, J. (2000).
1815:11858/00-001M-0000-002C-4314-2
1800:(3–4): 379–423 & 623–656.
1412:Error detection and correction
1015:List of error-correcting codes
370:adaptive modulation and coding
206:, while those that do not are
195:ECC is accomplished by adding
1:
2648:Digital Modulation and Coding
2522:MacWilliams, Florence Jessiem
2457:10.1016/S0019-9958(67)90835-2
1793:Bell System Technical Journal
1580:Bell System Technical Journal
1272:Reed–Solomon error correction
1216:Low-density parity-check code
961:Disadvantages of interleaving
929:aa_abbbbccccdddde_eef_ffg_gg
857:Interleaver designs include:
612:Low-density parity-check code
394:A block code (specifically a
2526:Sloane, Neil James Alexander
1529:"Overview of Channel Coding"
985:Software-defined radio (SDR)
897:or narrowband interference.
873:a contention-free quadratic
176:noisy-channel coding theorem
155:FEC information is added to
138:analog-to-digital conversion
84:way, most often by using an
2646:Wilson, Stephen G. (1996).
2623:Wicker, Stephen B. (1995).
1397:Burst error-correcting code
1071:5 (double-error correcting)
1048:3 (single-error correcting)
1038:3 (single-error correcting)
107:The American mathematician
2754:
2410:10.1038/s41598-017-11266-1
1770:Cambridge University Press
1258:Polar code (coding theory)
1231:, which is a near-optimal
1088:7 (three-error correcting)
1030:2 (single-error detecting)
895:frequency-selective fading
864:convolutional interleavers
799:
767:, e.g., for the design of
751:
679:
609:
575:
548:
379:
29:
2349:"Explaining Interleaving"
2203:10.1109/JPROC.2007.905132
1959:. ACM. pp. 106–115.
1843:10.1109/WCNC.2014.6952166
1682:Jim Cooke (August 2007).
1356:{\displaystyle 2^{n-1}-1}
1291:Triple modular redundancy
1042:Triple modular redundancy
309:triple modular redundancy
73:over unreliable or noisy
2318:, version 8.8.0, page 14
1527:Maunder, Robert (2016).
1427:Quantum error correction
1322:Cyclic redundancy checks
1166:is of practical interest
1051:perfect Hamming such as
879:3GPP Long Term Evolution
617:Low-density parity-check
527:forward error correction
329:and not at all below it.
65:is a technique used for
55:forward error correction
2484:10.1145/2070562.2070568
2472:"Rateless Spinal Codes"
2444:Information and Control
2295:10.1109/TIT.2007.896870
2190:Proceedings of the IEEE
2067:10.1145/1536414.1536422
2061:. ACM. pp. 39–44.
2024:10.1145/1326554.1326555
1872:3 February 2013 at the
1783:Shannon, C. E. (1948).
1575:Hamming, Richard Wesley
1305:, and the precursor to
1303:erasure correcting code
1243:Nordstrom-Robinson code
1222:, as the archetype for
1194:Latin square based code
1118:Algebraic geometry code
1082:Nordstrom-Robinson code
775:Locally decodable codes
472:Multidimensional parity
334:all-or-nothing tendency
146:soft-decision algorithm
2347:Techie (3 June 2010).
1644:29 August 2017 at the
1633:21 August 2016 at the
1377:
1357:
1282:Repeat-accumulate code
875:permutation polynomial
834:burst error-correction
817:communication channels
812:
780:Locally testable codes
754:Locally decodable code
430:to their block length.
411:
399:
75:communication channels
1975:10.1145/780542.780560
1889:; Viderman, Michael.
1432:Soft-decision decoder
1378:
1358:
1205:Linear Network Coding
1174:McEliece cryptosystem
810:
758:Locally testable code
405:
393:
327:signal-to-noise ratio
172:signal-to-noise ratio
140:in the receiver. The
90:error correcting code
86:error correction code
2717:error correction zoo
1837:. pp. 775–780.
1367:
1328:
1140:Constant-weight code
893:, e.g., to mitigate
830:uniform distribution
539:Levenshtein distance
2401:2017NatSR...711003K
2287:2006cs........1048T
2238:1995TDAPR.122...56D
2226:TDA Progress Report
1724:10.1155/2008/957846
1317:Walsh–Hadamard code
917:decoded incorrectly
802:optical interleaver
489:(SLC) NAND. Denser
448:Reed–Solomon coding
420:convolutional codes
32:optical interleaver
2560:(xxii+762+6 pages)
2379:Scientific Reports
2012:Journal of the ACM
1667:. Spansion. 2011.
1665:(Application note)
1373:
1353:
1224:sparse graph codes
1200:Lexicographic code
1145:Convolutional code
813:
655:10GBase-T Ethernet
628:Robert G. Gallager
412:
408:convolutional code
400:
386:Convolutional code
113:Hamming (7,4) code
67:controlling errors
47:information theory
2619:(x+2+208+4 pages)
2568:. New York, USA:
2547:978-0-444-85193-2
2336:. September 2009.
2197:(11): 2142–2156.
2136:978-0-7923-7868-6
2076:978-1-60558-506-2
1984:978-1-58113-674-6
1852:978-1-4799-3083-8
1376:{\displaystyle n}
1301:, a near-optimal
1265:, a near-optimal
1251:, a near-optimal
1164:Binary Golay code
1109:
1108:
1104:binary Golay code
1093:binary Golay code
1066:Extended Hamming
948:With interleaving
923:With interleaving
744:, respectively).
724:, and is sold by
487:single-level cell
435:Viterbi algorithm
294:
293:
223:Triplet received
131:cellular networks
71:data transmission
43:telecommunication
16:(Redirected from
2745:
2713:
2711:
2709:
2665:
2642:
2618:
2583:
2559:
2508:
2507:
2495:
2478:. pp. 1–6.
2467:
2461:
2460:
2459:
2450:(5–6): 613–616,
2439:
2433:
2432:
2422:
2412:
2394:
2370:
2364:
2363:
2361:
2359:
2344:
2338:
2337:
2325:
2319:
2313:
2307:
2306:
2280:
2271:(6): 2116–2132.
2258:
2252:
2251:
2249:
2221:
2215:
2214:
2184:
2178:
2177:
2165:
2159:
2158:
2147:
2141:
2140:
2118:
2109:
2108:
2102:
2098:
2096:
2088:
2050:
2044:
2043:
2003:
1997:
1996:
1968:
1966:quant-ph/0208062
1948:
1942:
1941:
1939:
1927:
1921:
1920:
1918:
1916:
1901:
1895:
1894:
1883:
1877:
1863:
1857:
1856:
1826:
1820:
1819:
1817:
1789:
1780:
1774:
1773:
1761:
1755:
1754:
1752:
1750:
1735:
1729:
1728:
1726:
1700:
1694:
1693:
1688:
1679:
1673:
1672:
1666:
1658:
1649:
1625:
1619:
1618:
1608:
1571:
1562:
1561:
1539:
1533:
1532:
1524:
1515:
1514:
1509:
1507:
1502:on 14 March 2012
1483:
1477:
1476:
1474:
1472:
1467:on 14 March 2012
1448:
1382:
1380:
1379:
1374:
1362:
1360:
1359:
1354:
1346:
1345:
1287:Repetition codes
1277:Reed–Muller code
1218:, also known as
1184:Hagelbarger code
1019:
742:Mobile Broadband
734:Broadband Access
726:Verizon Wireless
710:Verizon Wireless
621:channel capacity
531:Hamming distance
491:multi-level cell
460:hard disk drives
220:
21:
2753:
2752:
2748:
2747:
2746:
2744:
2743:
2742:
2728:
2727:
2707:
2705:
2700:
2697:
2662:
2645:
2639:
2622:
2607:
2586:
2580:
2563:
2548:
2520:
2517:
2515:Further reading
2512:
2511:
2504:
2469:
2468:
2464:
2441:
2440:
2436:
2372:
2371:
2367:
2357:
2355:
2346:
2345:
2341:
2327:
2326:
2322:
2314:
2310:
2260:
2259:
2255:
2247:10.1.1.105.6640
2223:
2222:
2218:
2186:
2185:
2181:
2167:
2166:
2162:
2149:
2148:
2144:
2137:
2127:Springer Verlag
2120:
2119:
2112:
2099:
2089:
2077:
2052:
2051:
2047:
2005:
2004:
2000:
1985:
1950:
1949:
1945:
1929:
1928:
1924:
1914:
1912:
1910:Quanta Magazine
1903:
1902:
1898:
1885:
1884:
1880:
1874:Wayback Machine
1864:
1860:
1853:
1828:
1827:
1823:
1787:
1782:
1781:
1777:
1763:
1762:
1758:
1748:
1746:
1737:
1736:
1732:
1702:
1701:
1697:
1686:
1681:
1680:
1676:
1664:
1660:
1659:
1652:
1646:Wayback Machine
1635:Wayback Machine
1626:
1622:
1573:
1572:
1565:
1558:
1541:
1540:
1536:
1526:
1525:
1518:
1505:
1503:
1485:
1484:
1480:
1470:
1468:
1450:
1449:
1445:
1440:
1393:
1365:
1364:
1331:
1326:
1325:
1162:, of which the
1017:
976:
963:
955:
942:
930:
912:
903:
805:
798:
760:
752:Main articles:
750:
684:
678:
614:
608:
580:
574:
553:
547:
428:polynomial time
388:
380:Main articles:
378:
318:
266:1 (error-free)
234:0 (error-free)
226:Interpreted as
215:repetition code
193:
164:computer memory
142:Viterbi decoder
109:Richard Hamming
102:reverse channel
35:
28:
23:
22:
15:
12:
11:
5:
2751:
2749:
2741:
2740:
2730:
2729:
2726:
2725:
2720:
2714:
2696:
2695:External links
2693:
2692:
2691:
2684:
2678:
2672:
2666:
2660:
2643:
2637:
2620:
2605:
2584:
2578:
2561:
2546:
2516:
2513:
2510:
2509:
2502:
2462:
2434:
2365:
2353:W3 Techie Blog
2339:
2320:
2316:3GPP TS 36.212
2308:
2253:
2216:
2179:
2160:
2142:
2135:
2110:
2101:|journal=
2075:
2045:
1998:
1983:
1943:
1922:
1896:
1878:
1867:"802.11n-2009"
1858:
1851:
1821:
1775:
1756:
1730:
1695:
1689:. p. 28.
1674:
1650:
1620:
1563:
1556:
1534:
1516:
1478:
1442:
1441:
1439:
1436:
1435:
1434:
1429:
1424:
1419:
1414:
1409:
1404:
1399:
1392:
1389:
1388:
1387:
1372:
1352:
1349:
1344:
1341:
1338:
1334:
1319:
1314:
1309:
1307:Fountain codes
1296:
1293:
1284:
1279:
1274:
1269:
1260:
1255:
1246:
1240:
1235:
1226:
1213:
1208:
1202:
1197:
1191:
1186:
1181:
1176:
1172:, used in the
1167:
1157:
1152:
1150:Expander codes
1147:
1142:
1137:
1132:
1126:
1120:
1115:
1107:
1106:
1100:
1096:
1095:
1089:
1085:
1084:
1079:
1075:
1074:
1072:
1068:
1067:
1064:
1056:
1055:
1049:
1045:
1044:
1039:
1035:
1034:
1031:
1027:
1026:
1023:
1016:
1013:
1012:
1011:
1005:
999:
975:
972:
968:neural network
962:
959:
952:
940:
927:
910:
902:
899:
883:
882:
871:
868:
865:
862:
797:
794:
790:subexponential
749:
746:
680:Main article:
677:
674:
670:fountain codes
657:(802.3an) and
610:Main article:
607:
604:
576:Main article:
573:
570:
546:
543:
535:bit error rate
444:
443:
431:
377:
374:
358:bit error rate
350:
349:
345:
330:
317:
314:
305:
304:
301:
292:
291:
288:
284:
283:
280:
276:
275:
272:
268:
267:
264:
260:
259:
256:
252:
251:
248:
244:
243:
240:
236:
235:
232:
228:
227:
224:
208:non-systematic
192:
189:
180:Claude Shannon
150:bit-error rate
63:channel coding
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2750:
2739:
2736:
2735:
2733:
2724:
2721:
2718:
2715:
2704:
2699:
2698:
2694:
2689:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2663:
2661:0-13-210071-1
2657:
2653:
2652:Prentice-Hall
2649:
2644:
2640:
2638:0-13-200809-2
2634:
2630:
2629:Prentice-Hall
2626:
2621:
2616:
2612:
2608:
2606:0-262-01098-4
2602:
2598:
2594:
2590:
2585:
2581:
2579:0-306-40615-2
2575:
2571:
2567:
2562:
2557:
2553:
2549:
2543:
2539:
2535:
2534:North-Holland
2531:
2527:
2523:
2519:
2518:
2514:
2505:
2503:9781450310598
2499:
2494:
2489:
2485:
2481:
2477:
2473:
2466:
2463:
2458:
2453:
2449:
2445:
2438:
2435:
2430:
2426:
2421:
2416:
2411:
2406:
2402:
2398:
2393:
2388:
2384:
2380:
2376:
2369:
2366:
2354:
2350:
2343:
2340:
2335:
2331:
2324:
2321:
2317:
2312:
2309:
2304:
2300:
2296:
2292:
2288:
2284:
2279:
2274:
2270:
2266:
2265:
2257:
2254:
2248:
2243:
2239:
2235:
2231:
2227:
2220:
2217:
2212:
2208:
2204:
2200:
2196:
2192:
2191:
2183:
2180:
2176:. April 2009.
2175:
2171:
2164:
2161:
2156:
2152:
2151:Luby, Michael
2146:
2143:
2138:
2132:
2128:
2124:
2117:
2115:
2111:
2106:
2094:
2086:
2082:
2078:
2072:
2068:
2064:
2060:
2056:
2049:
2046:
2041:
2037:
2033:
2029:
2025:
2021:
2017:
2013:
2009:
2002:
1999:
1994:
1990:
1986:
1980:
1976:
1972:
1967:
1962:
1958:
1954:
1947:
1944:
1938:
1933:
1926:
1923:
1911:
1907:
1900:
1897:
1892:
1888:
1887:Kaufman, Tali
1882:
1879:
1875:
1871:
1868:
1862:
1859:
1854:
1848:
1844:
1840:
1836:
1832:
1825:
1822:
1816:
1811:
1807:
1803:
1799:
1795:
1794:
1786:
1779:
1776:
1771:
1767:
1760:
1757:
1745:
1741:
1734:
1731:
1725:
1720:
1716:
1712:
1711:
1706:
1699:
1696:
1692:
1685:
1678:
1675:
1671:
1663:
1657:
1655:
1651:
1647:
1643:
1640:
1636:
1632:
1629:
1624:
1621:
1616:
1612:
1607:
1602:
1598:
1594:
1590:
1586:
1582:
1581:
1576:
1570:
1568:
1564:
1559:
1557:0-927239-00-0
1553:
1549:
1545:
1538:
1535:
1530:
1523:
1521:
1517:
1513:
1501:
1497:
1493:
1489:
1482:
1479:
1466:
1462:
1458:
1454:
1447:
1444:
1437:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1413:
1410:
1408:
1407:Erasure codes
1405:
1403:
1400:
1398:
1395:
1394:
1390:
1386:
1370:
1350:
1347:
1342:
1339:
1336:
1332:
1323:
1320:
1318:
1315:
1313:
1310:
1308:
1304:
1300:
1297:
1294:
1292:
1288:
1285:
1283:
1280:
1278:
1275:
1273:
1270:
1268:
1264:
1261:
1259:
1256:
1254:
1250:
1247:
1244:
1241:
1239:
1236:
1234:
1230:
1227:
1225:
1221:
1220:Gallager code
1217:
1214:
1212:
1209:
1206:
1203:
1201:
1198:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1179:Hadamard code
1177:
1175:
1171:
1168:
1165:
1161:
1158:
1156:
1153:
1151:
1148:
1146:
1143:
1141:
1138:
1136:
1133:
1130:
1127:
1124:
1121:
1119:
1116:
1114:
1111:
1110:
1105:
1101:
1098:
1097:
1094:
1090:
1087:
1086:
1083:
1080:
1077:
1076:
1073:
1070:
1069:
1065:
1062:
1058:
1057:
1054:
1050:
1047:
1046:
1043:
1040:
1037:
1036:
1032:
1029:
1028:
1024:
1021:
1020:
1014:
1009:
1006:
1003:
1000:
997:
994:
993:
992:
989:
986:
982:
973:
971:
969:
960:
958:
951:
949:
945:
939:
937:
933:
926:
924:
920:
918:
909:
907:
900:
898:
896:
892:
888:
880:
876:
872:
869:
866:
863:
860:
859:
858:
855:
853:
848:
846:
842:
837:
835:
831:
826:
822:
818:
809:
803:
795:
793:
791:
786:
783:
781:
776:
772:
770:
766:
759:
755:
747:
745:
743:
739:
735:
731:
727:
723:
719:
715:
711:
707:
703:
698:
696:
692:
691:Shannon limit
688:
683:
675:
673:
671:
667:
664:
660:
656:
652:
648:
644:
640:
635:
633:
629:
624:
622:
618:
613:
605:
603:
601:
597:
593:
588:
585:
579:
571:
569:
565:
561:
557:
552:
544:
542:
540:
536:
532:
528:
523:
521:
516:
514:
513:finite fields
509:
507:
503:
502:soft-decision
499:
498:hard-decision
494:
492:
488:
484:
479:
477:
476:Hamming codes
473:
469:
465:
461:
457:
453:
452:compact discs
449:
440:
439:exponentially
436:
432:
429:
425:
424:
423:
421:
417:
409:
406:A continuous
404:
397:
392:
387:
383:
375:
373:
371:
367:
363:
359:
355:
346:
343:
342:Shannon limit
339:
335:
331:
328:
324:
323:
322:
315:
313:
310:
302:
299:
298:
297:
289:
286:
285:
281:
278:
277:
273:
270:
269:
265:
262:
261:
257:
254:
253:
249:
246:
245:
241:
238:
237:
233:
230:
229:
225:
222:
221:
218:
216:
211:
209:
205:
204:
198:
190:
188:
186:
181:
177:
173:
167:
165:
162:
158:
153:
151:
147:
144:implements a
143:
139:
134:
132:
128:
123:
121:
116:
114:
110:
105:
103:
99:
98:detect errors
95:
91:
87:
83:
78:
76:
72:
68:
64:
60:
56:
52:
51:coding theory
48:
44:
40:
33:
19:
2706:. Retrieved
2647:
2624:
2588:
2570:Plenum Press
2565:
2529:
2493:1721.1/79676
2475:
2465:
2447:
2443:
2437:
2385:(1): 11003.
2382:
2378:
2368:
2356:. Retrieved
2352:
2342:
2329:
2323:
2311:
2268:
2262:
2256:
2229:
2225:
2219:
2194:
2188:
2182:
2169:
2163:
2154:
2145:
2122:
2058:
2048:
2015:
2011:
2001:
1956:
1946:
1925:
1913:. Retrieved
1909:
1899:
1881:
1861:
1834:
1824:
1797:
1791:
1778:
1765:
1759:
1747:. Retrieved
1743:
1733:
1714:
1708:
1698:
1690:
1677:
1668:
1623:
1584:
1578:
1548:Cirrus Logic
1543:
1537:
1511:
1504:. Retrieved
1500:the original
1495:
1491:
1481:
1469:. Retrieved
1465:the original
1460:
1456:
1446:
1299:Tornado code
1238:m of n codes
1189:Hamming code
1053:Hamming(7,4)
990:
977:
970:structures.
964:
956:
947:
946:
943:
935:
934:
931:
922:
921:
913:
905:
904:
884:
856:
852:factor graph
849:
838:
814:
796:Interleaving
787:
784:
773:
761:
741:
738:Power Vision
737:
733:
708:and sold by
699:
693:. Predating
687:Turbo coding
685:
651:IEEE 802.11n
647:IEEE 802.16e
636:
632:Reed–Solomon
625:
615:
589:
584:concatenated
583:
581:
566:
562:
558:
554:
524:
517:
510:
501:
497:
495:
480:
445:
413:
396:Hamming code
354:channel code
351:
338:cliff effect
333:
319:
306:
295:
212:
207:
201:
194:
168:
157:mass storage
154:
135:
124:
117:
106:
93:
89:
85:
79:
62:
58:
54:
36:
2538:Elsevier BV
2018:(1): 1–16.
1749:20 December
1606:10945/46756
1591:: 147–160.
1422:Linear code
1263:Raptor code
1249:Online code
1160:Golay codes
1155:Group codes
1135:Berger code
1129:Barker code
841:turbo codes
702:CDMA2000 1x
676:Turbo codes
659:G.hn/G.9960
416:block codes
18:Interleaver
2677:2004-11-29
2671:2007-02-16
2392:1705.09334
2332:(V1.1.1).
2330:En 302 755
2278:cs/0601048
2232:: 42–122.
2172:(V1.2.1).
2170:En 302 307
1937:2311.00558
1587:(2). USA:
1438:References
1312:Turbo code
1289:, such as
1170:Goppa code
1099:8 (TECFED)
845:LDPC codes
695:LDPC codes
682:Turbo code
549:See also:
520:LDPC codes
483:NAND flash
382:Block code
203:systematic
197:redundancy
185:polar code
2593:MIT Press
2242:CiteSeerX
2103:ignored (
2093:cite book
2085:263865692
2032:0004-5411
1915:9 January
1492:Crosslink
1457:Crosslink
1402:Code rate
1348:−
1340:−
1211:Long code
1102:extended
891:diversity
885:In multi-
825:code word
792:lengths.
592:Voyager 2
120:multicast
82:redundant
39:computing
2732:Category
2615:87-21889
2556:76-41296
2429:28887480
2040:14617710
1993:10585919
1870:Archived
1717:: 1–12.
1642:Archived
1631:Archived
1615:61141773
1589:AT&T
1391:See also
1123:BCH code
1113:AN codes
1091:perfect
1022:Distance
722:Qualcomm
706:Qualcomm
2708:5 March
2420:5591216
2397:Bibcode
2283:Bibcode
2234:Bibcode
2211:9289140
1506:5 March
1471:5 March
1229:LT code
1033:Parity
1008:OpenAir
901:Example
887:carrier
718:1xEV-DO
600:Galileo
129:and in
2658:
2635:
2613:
2603:
2576:
2554:
2544:
2500:
2427:
2417:
2358:3 June
2301:
2244:
2209:
2133:
2083:
2073:
2038:
2030:
1991:
1981:
1849:
1744:USENIX
1670:flash.
1613:
1554:
1383:, see
1061:SECDED
996:AFF3CT
821:bursts
730:Sprint
714:Sprint
639:DVB-S2
598:. The
596:Uranus
474:, and
458:, and
336:– the
191:Method
174:. The
127:modems
49:, and
2387:arXiv
2299:S2CID
2273:arXiv
2207:S2CID
2081:S2CID
2036:S2CID
1989:S2CID
1961:arXiv
1932:arXiv
1788:(PDF)
1687:(PDF)
1611:S2CID
1025:Code
983:in a
668:(see
643:WiMAX
525:Most
464:Golay
376:Types
348:data.
332:This
61:) or
2710:2006
2656:ISBN
2633:ISBN
2611:LCCN
2601:ISBN
2574:ISBN
2552:LCCN
2542:ISBN
2498:ISBN
2425:PMID
2360:2010
2334:ETSI
2174:ETSI
2131:ISBN
2105:help
2071:ISBN
2028:ISSN
1979:ISBN
1917:2024
1847:ISBN
1772:, UK
1751:2018
1715:2008
1552:ISBN
1508:2006
1473:2006
1002:IT++
843:and
756:and
740:and
666:MBMS
663:3GPP
506:BCJR
456:DVDs
418:and
384:and
287:011
279:101
271:110
263:111
255:100
247:010
239:001
231:000
2488:hdl
2480:doi
2452:doi
2415:PMC
2405:doi
2303:660
2291:doi
2230:122
2199:doi
2063:doi
2020:doi
1971:doi
1839:doi
1810:hdl
1802:doi
1719:doi
1601:hdl
1593:doi
1059:4 (
672:).
653:),
468:BCH
366:ARQ
178:of
161:ECC
94:ECC
88:or
69:in
59:FEC
37:In
2734::
2654:.
2631:.
2609:.
2599:.
2591:.
2572:.
2550:.
2540:.
2536:/
2524:;
2496:.
2486:.
2474:.
2448:11
2446:,
2423:.
2413:.
2403:.
2395:.
2381:.
2377:.
2351:.
2297:.
2289:.
2281:.
2269:53
2267:.
2240:.
2228:.
2205:.
2195:95
2193:.
2129:.
2125:.
2113:^
2097::
2095:}}
2091:{{
2079:.
2069:.
2057:.
2034:.
2026:.
2016:55
2014:.
2010:.
1987:.
1977:.
1969:.
1955:.
1908:.
1845:.
1833:.
1808:.
1798:27
1796:.
1790:.
1768:,
1742:.
1713:.
1707:.
1653:^
1609:.
1599:.
1585:29
1583:.
1566:^
1550:.
1519:^
1510:.
1494:.
1490:.
1459:.
1455:.
950::
938::
925::
919:.
908::
836:.
771:.
728:,
712:,
478:.
470:,
466:,
454:,
422:.
290:1
282:1
274:1
258:0
250:0
242:0
210:.
133:.
122:.
115:.
77:.
53:,
45:,
41:,
2712:.
2664:.
2641:.
2617:.
2582:.
2558:.
2506:.
2490::
2482::
2454::
2431:.
2407::
2399::
2389::
2383:7
2362:.
2305:.
2293::
2285::
2275::
2250:.
2236::
2213:.
2201::
2157:.
2139:.
2107:)
2087:.
2065::
2042:.
2022::
1995:.
1973::
1963::
1940:.
1934::
1919:.
1893:.
1855:.
1841::
1818:.
1812::
1804::
1753:.
1727:.
1721::
1617:.
1603::
1595::
1560:.
1531:.
1496:3
1475:.
1461:3
1371:n
1351:1
1343:1
1337:n
1333:2
1063:)
804:.
645:(
344:.
92:(
57:(
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.