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# Forward error correction  ( Error Detection And Correction )

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In telecommunication, information theory, and , forward error correction ( FEC) or channel coding is a technique used for in data transmission over unreliable or noisy communication channels. The central idea is the sender encodes the message in a redundant way by using an error-correcting code (ECC). The American mathematician pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code.

The redundancy allows the receiver to detect a limited number of errors that may occur anywhere in the message, and often to correct these errors without retransmission. FEC gives the receiver the ability to correct errors without needing a to request retransmission of data, but at the cost of a fixed, higher forward channel bandwidth. FEC is therefore applied in situations where retransmissions are costly or impossible, such as one-way communication links and when transmitting to multiple receivers in . For example, in the case of a satellite orbiting around , a retransmission because of decoding errors can create a delay of 5 hours. FEC information is usually added to devices to enable recovery of corrupted data, is widely used in , and is used on systems where the primary memory is .

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 analog-to-digital conversion in the receiver. The implements a soft-decision algorithm to demodulate digital data from an analog signal corrupted by noise. Many FEC coders can also generate a (BER) signal which can be used as feedback to fine-tune the analog receiving electronics.

The maximum fractions of errors or of missing bits that can be corrected is determined by the design of the FEC code, 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 bandwidth, which reduces the effective bit-rate while improving the received effective signal-to-noise ratio. The noisy-channel coding theorem of Claude Shannon answers the question of how much bandwidth is left for data communication while using the most efficient code that turns the decoding error probability to zero. 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 no insight of how to build a capacity achieving code. Fortunately, after years of research, some advanced FEC systems nowadays come very close to the theoretical maximum.

How it works
FEC is accomplished by adding redundancy to the transmitted information using an algorithm. A redundant bit may be a complex 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 , while those that do not are non-systematic.

A simplistic example of FEC is to transmit each data bit 3 times, which is known as a (3,1) . Through a noisy channel, a receiver might see 8 versions of the output, see table below.

0000 (error free)
0010
0100
1000
1111 (error free)
1101
1011
0111
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 FEC is:
• Up to 1 bit of triplet in error, or
• up to 2 bits of triplet omitted (cases not shown in table).
Though simple to implement and widely used, this triple modular redundancy is a relatively inefficient FEC. Better FEC codes typically examine the last several dozen, or even the last several hundred, previously received bits to determine how to decode the current small handful of bits (typically in groups of 2 to 8 bits).

Averaging noise to reduce errors
FEC 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.
• Because of this "risk-pooling" effect, digital communication systems that use FEC tend to work well above a certain minimum signal-to-noise ratio and not at all below it.
• This all-or-nothing tendency — the — becomes more pronounced as stronger codes are used that more closely approach the theoretical .
• Interleaving FEC coded data can reduce the all or nothing properties of transmitted FEC codes when the channel errors tend to occur in bursts. However, this method has limits; it is best used on narrowband data.

Most telecommunication systems use a fixed designed to tolerate the expected worst-case bit error rate, 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 hybrid automatic repeat-request use a fixed FEC method as long as the FEC can handle the error rate, then switch to ARQ when the error rate gets too high; adaptive modulation and coding uses a variety of FEC 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.

Types of FEC
The two main categories of FEC codes are and convolutional codes.
• Block codes work on fixed-size blocks (packets) of bits or symbols of predetermined size. Practical block codes can generally be hard-decoded in to their block length.
• Convolutional codes work on bit or symbol streams of arbitrary length. They are most often soft decoded with the Viterbi algorithm, 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 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 convolutional codes include "tail-biting" and "bit-flushing".

There are many types of block codes, but among the classical ones the most notable is Reed-Solomon coding because of its widespread use in , , and hard disk drives. Other examples of classical block codes include Golay, , Multidimensional parity, and .

Hamming ECC is commonly used to correct memory errors. "Hamming codes for NAND flash memory devices". EE Times-Asia. Apparently based on "Micron Technical Note TN-29-08: Hamming Codes for NAND Flash Memory Devices". 2005. Both say: "The Hamming algorithm is an industry-accepted method for error detection and correction in many SLC NAND flash-based applications." This provides single-bit error correction and 2-bit error detection. Hamming codes are only suitable for more reliable single level cell (SLC) NAND. Denser multi level cell (MLC) NAND requires stronger multi-bit correcting ECC such as BCH or Reed–Solomon. "What Types of ECC Should Be Used on Flash Memory?". (Spansion application note). 2011. says: "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 flash." Jim Cooke. "The Inconvenient Truths of NAND Flash Memory". 2007. p. 28. says "For SLC, a code with a correction threshold of 1 is sufficient. t=4 required ... for MLC."

NOR Flash typically does not use any error correction.

Classical block codes are usually decoded using hard-decision 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 soft-decision algorithms like the Viterbi, MAP or algorithms, which process (discretized) analog signals, and which allow for much higher error-correction performance than hard-decision decoding.

Nearly all classical block codes apply the algebraic properties of . Hence classical block codes are often referred to as algebraic codes.

In contrast to classical block codes that often specify an error-detecting or error-correcting ability, many modern block codes such as lack such guarantees. Instead, modern codes are evaluated in terms of their bit error rates.

Most forward error correction codes correct only bit-flips, but not bit-insertions or bit-deletions. In this setting, the is the appropriate way to measure the bit error rate. A few forward error correction codes are designed to correct bit-insertions and bit-deletions, such as and . The Levenshtein distance is a more appropriate way to measure the bit error rate when using such codes.

Code-rate and the tradeoff between reliability and data rate
The fundamental principle of FEC 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 FEC system is defined as the rate between the number of information bits and the total number of bits (i.e. information 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.

Interestingly, 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 the receiver SNR decreasing the bit error rate, at the cost of reducing the effective data rate. On the other extreme, not using any FEC (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.

One interesting question is the following: how efficient in terms of information transfer can be a FEC 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 FEC whose error rate tends to zero:C. E. Shannon: A mathematical theory of communication. Bell System Technical Journal, vol. 27, pp. 379–423 and 623–656, July and October 1948. His proof, unfortunately, relies on Gaussian random coding, which is not suitable of real-world applications. This upper bound given by Shannon's work set up a long journey in designing FECs that can go close to the ultimate performance boundary. Various codes today can attain almost the Shannon limit. However, capacity achieving FECs are usually extremely complex to implement.

The most popular codes FECs have a trade performance and computational complexity. Usually their parameters give a range of possible code rates, which can be optimized depending of the scenario. Usually, this optimization is done in order to achieve a low decoding error probability without hurting too much the data rate. Another criteria for optimizing the code rate is to balance low error rate and retransmissions number in order to the energy cost of the communication .

Concatenated FEC codes for improved performance
Classical (algebraic) block codes and convolutional codes are frequently combined in concatenated 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 codes can yield very low error rates, but for long range transmission conditions (like deep space) iterative decoding is recommended.

Concatenated codes have been standard practice in satellite and deep space communications since first used the technique in its 1986 encounter with . The Galileo craft used iterative concatenated codes to compensate for the very high error rate conditions caused by having a failed antenna.

Low-density parity-check (LDPC)
Low-density parity-check (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 (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.

LDPC codes were first introduced by Robert G. Gallager in his PhD thesis in 1960, but due to the computational effort in implementing encoder and decoder and the introduction of Reed–Solomon codes, they were mostly ignored until recently.

LDPC codes are now used in many recent high-speed communication standards, such as DVB-S2 (Digital video broadcasting), (IEEE 802.16e standard for microwave communications), High-Speed Wireless LAN (IEEE 802.11n),IEEE Standard, section 20.3.11.6 "802.11n-2009", IEEE, October 29, 2009, accessed March 21, 2011. 10GBase-T Ethernet (802.3an) and G.hn/G.9960 (ITU-T Standard for networking over power lines, phone lines and coaxial cable). Other LDPC codes are standardized for wireless communication standards within 3GPP (see fountain codes).

Turbo codes
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 . Predating in terms of practical application, they now provide similar performance.

One of the earliest commercial applications of turbo coding was the CDMA2000 1x (TIA IS-2000) digital cellular technology developed by and sold by , , and other carriers. It is also used for the evolution of CDMA2000 1x specifically for Internet access, 1xEV-DO (TIA IS-856). Like 1x, EV-DO was developed by , and is sold by , , and other carriers (Verizon's marketing name for 1xEV-DO is Broadband Access, Sprint's consumer and business marketing names for 1xEV-DO are Power Vision and Mobile Broadband, respectively).

Local decoding and testing of codes
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 bits of the message are of interest for now. Also such codes have become an important tool in computational complexity theory, e.g., for the design of probabilistically checkable proofs.

Locally decodable codes 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. Locally testable codes 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.

Interleaving
Interleaving is frequently used in digital communication and storage systems to improve the performance of forward error correcting codes. Many communication channels are not memoryless: errors typically occur in rather than independently. If the number of errors within a exceeds the error-correcting code's capability, it fails to recover the original code word. Interleaving ameliorates this problem by shuffling source symbols across several code words, thereby creating a more uniform distribution of errors.
(2018). 9780792378686, .
Therefore, interleaving is widely used for burst error-correction.

The analysis of modern iterated codes, like and , typically assumes an independent distribution of errors. Systems using LDPC codes therefore typically employ additional interleaving across the symbols within a code word.

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 that represents the decoder; the interleaver is chosen to avoid short cycles.

Interleaver designs include:

• rectangular (or uniform) interleavers (similar to the method using skip factors described above)
• convolutional interleavers
• random interleavers (where the interleaver is a known random permutation)
• 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).S. Dolinar and D. Divsalar. Weight Distributions for Turbo Codes Using Random and Nonrandom Permutations. 1995. [6]
• Another possible construction is a contention-free quadratic permutation polynomial (QPP). It is used for example in the 3GPP Long Term Evolution mobile telecommunication standard. 3GPP TS 36.212, version 8.8.0, page 14

In multi- communication systems, interleaving across carriers may be employed to provide frequency , e.g., to mitigate frequency-selective fading or narrowband interference.

Example
Transmission without interleaving:

```Error-free message:                                 aaaabbbbccccddddeeeeffffgggg
Transmission with a burst error:                    aaaabbbbccc____deeeeffffgggg
```

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 .

With interleaving:

```Error-free code words:                              aaaabbbbccccddddeeeeffffgggg
Interleaved:                                        abcdefgabcdefgabcdefgabcdefg
Transmission with a burst error:                    abcdefgabcd____bcdefgabcdefg
Received code words after deinterleaving:           aa_abbbbccccdddde_eef_ffg_gg
```

In each of the codewords aaaa, eeee, ffff, gggg, only one bit is altered, so one-bit error-correcting code will decode everything correctly.

Transmission without interleaving:

```Original transmitted sentence:                      ThisIsAnExampleOfInterleaving
Received sentence with a burst error:               ThisIs______pleOfInterleaving
```

The term "AnExample" ends up mostly unintelligible and difficult to correct.

With interleaving:

```Transmitted sentence:                               ThisIsAnExampleOfInterleaving...
Error-free transmission:                            TIEpfeaghsxlIrv.iAaenli.snmOten.
Received sentence with a burst error:               TIEpfe______Irv.iAaenli.snmOten.
```

No word is completely lost and the missing letters can be recovered with minimal guesswork.

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 and achieve more reliable communication than a simpler decoder combined with an interleaver.

Software for error-correcting codes
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 in a Software-defined radio (SDR) 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 energy efficiency of the system.

In this context, there are various available Open-source software listed below (non exhaustive).

• AFF3CT(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).
• IT++: a C++ library of classes and functions for linear algebra, numerical optimization, signal processing, communications, and statistics.
• OpenAir: implementation (in C) of the 3GPP specifications concerning the Evolved Packet Core Networks.

List of error-correcting codes
Parity
Triple modular redundancy
perfect Hamming such as Hamming(7,4)
Extended Hamming
perfect binary Golay code
extended binary Golay code

• , which can be designed to correct any arbitrary number of errors per code block.
• Constant-weight code
• Convolutional code
• Golay codes, of which the Binary Golay code is of practical interest
• , used in the McEliece cryptosystem
• Latin square based code for non-white noise (prevalent for example in broadband over powerlines)
• Lexicographic code
• Long code
• Low-density parity-check code, also known as , as the archetype for sparse graph codes
• , which is a near-optimal
• m of n codes
• , a near-optimal
• Polar code (coding theory)
• , a near-optimal
• Reed–Solomon error correction
• Reed–Muller code
• Repeat-accumulate code
• , such as Triple modular redundancy
• Spinal code, a rateless, nonlinear code based on pseudo-random hash functions
• , a near-optimal , and the precursor to
• Cyclic redundancy checks (CRCs) can correct 1-bit errors for messages at most $2^\left\{n-1\right\}-1$ bits long for optimal generator polynomials of degree $n$, see Mathematics of cyclic redundancy checks#Bitfilters

• Soft-decision decoder
• Error detection and correction
• Error-correcting codes with feedback
• Burst error-correcting code

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