Error correction code

Simplified formalism

Formally, an error-correcting code is given by its (injective) encoding function ${\displaystyle f:\Sigma ^{*}\hookrightarrow \Delta ^{*}}$ which assigns to each word ${\displaystyle u\in \Sigma ^{*}\;{\stackrel {\mathrm {def} }{=}}\;\textstyle \bigcup _{n\in \mathbb {N} }\Sigma ^{n}}$ of a finite alphabet ${\displaystyle \Sigma }$ a unique word ${\displaystyle w\in \Delta ^{*}\;{\stackrel {\mathrm {def} }{=}}\;\textstyle \bigcup _{n\in \mathbb {N} }\Delta ^{n}}$ (a concatenation of letters) from the alphabet ${\displaystyle \Delta }$.

Most commonly, ${\displaystyle f}$ is a homomorphism in the sense that if ${\displaystyle x_{1}*x_{2}}$ is the concatenation of ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$, then we have the following:$${\displaystyle f(u_{1}*u_{2})=f(u_{1})*f(u_{2})}$$This implies that it is enough to define ${\displaystyle f(\sigma )}$ for single-letter words ${\displaystyle \sigma }$. The range ${\displaystyle \mathrm {Im} (f)}$ of the function ${\displaystyle f}$ is the set of code-words. The capabilities of the code to detect and correct errors can then be understood from the distance ${\displaystyle d=\min _{w_{1},w_{2}\in \mathrm {Im} (f);\ w_{1}\neq w_{2}}d_{H}(w_{1},w_{2})}$ of the code, which is the minimum Hamming distance separating any two distinct code words. A code with distance ${\displaystyle d}$ can detect errors on ${\displaystyle k}$ bits as long as ${\displaystyle k<d}$, and among those detected errors, the code can correct ${\displaystyle l}$-bit errors whenever ${\displaystyle 2l<d}$.

Averaging noise to reduce errors

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.

• Because of this "risk-pooling" effect, digital communication systems that use ECC tend to work well above a certain minimum signal-to-noise ratio and not at all below it.
• This all-or-nothing tendency – the cliff effect – becomes more pronounced as stronger codes are used that more closely approach the theoretical Shannon limit.
• 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 data.

Most telecommunication systems use a fixed channel code 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 ECC method as long as the ECC can handle the error rate, then switch to ARQ when the error rate gets too high; adaptive modulation and coding 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.

Block codes

There are many types of block codes; Reed–Solomon coding is noteworthy for its widespread use in compact discs, DVDs, and hard disk drives. Other examples of classical block codes include Golay, BCH, Multidimensional parity, and Hamming codes.

Hamming ECC is commonly used to correct ECC memory and early SLC NAND flash memory errors.^[6] 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 may use multi-bit correcting ECC such as BCH, Reed–Solomon, or LDPC.^[7][8] NOR flash typically does not use any error correction.^[7]

Soft codes

Code-rate and the tradeoff between reliability and data rate

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 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.

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.^[11] In one extreme, a strong code (with low code-rate) can induce an important increase in 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.

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 whose error rate tends to zero:^[12] 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.

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 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.^[13]

Concatenation (combination)

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 Voyager 2 first used the technique in its 1986 encounter with Uranus. The Galileo craft used iterative concatenated codes to compensate for the very high error rate conditions caused by having a failed antenna.

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 bursts rather than independently. If the number of errors within a code word 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 uniform distribution of errors.^[18] Therefore, interleaving is widely used for burst error-correction.

The analysis of modern iterated codes, like turbo codes and LDPC codes, typically assumes an independent distribution of errors.^[19] Systems using LDPC codes therefore typically employ additional interleaving across the symbols within a code word.^[20]

For turbo codes, an interleaver is an integral component and its proper design is crucial for good performance.^[18][21] The iterative decoding algorithm works best when there are not short cycles in the factor graph 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).^[22]
• a contention-free quadratic permutation polynomial (QPP).^[23] An example of use is in the 3GPP Long Term Evolution mobile telecommunication standard.^[24]

In multi-carrier communication systems, interleaving across carriers may be employed to provide frequency diversity, e.g., to mitigate frequency-selective fading or narrowband interference.^[25]

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

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

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

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