Error correction by detection and repetition

In information theory and coding theory with applications in computer science and telecommunication, error detection and correction (EDAC) or error control are techniques that enable reliable delivery of digital data over unreliable communication channels. Many communication channels are subject to channel noise, and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases.

Definitions[edit]

Error detection is the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver.

Error correction is the detection of errors and reconstruction of the original, error-free data.

History[edit]

In classical antiquity, copyists of the Hebrew Bible were paid for their work according to the number of stichs (lines of verse). As the prose books of the Bible were hardly ever written in stichs, the copyists, in order to estimate the amount of work, had to count the letters.^[1] This also helped ensure accuracy in the transmission of the text with the production of subsequent copies.^[2][3] Between the 7th and 10th centuries CE a group of Jewish scribes formalized and expanded this to create the Numerical Masorah to ensure accurate reproduction of the sacred text. It included counts of the number of words in a line, section, book and groups of books, noting the middle stich of a book, word use statistics, and commentary.^[1] Standards became such that a deviation in even a single letter in a Torah scroll was considered unacceptable.^[4] The effectiveness of their error correction method was verified by the accuracy of copying through the centuries demonstrated by discovery of the Dead Sea Scrolls in 1947–1956, dating from c.150 BCE-75 CE.^[5]

The modern development of error correction codes is credited to Richard Hamming in 1947.^[6] A description of Hamming’s code appeared in Claude Shannon’s A Mathematical Theory of Communication^[7] and was quickly generalized by Marcel J. E. Golay.^[8]

Introduction[edit]

All error-detection and correction schemes add some redundancy (i.e., some extra data) to a message, which receivers can use to check consistency of the delivered message, and to recover data that has been determined to be corrupted. Error-detection and correction schemes can be either systematic or non-systematic. In a systematic scheme, the transmitter sends the original data, and attaches a fixed number of check bits (or parity data), which are derived from the data bits by some deterministic algorithm. If only error detection is required, a receiver can simply apply the same algorithm to the received data bits and compare its output with the received check bits; if the values do not match, an error has occurred at some point during the transmission. In a system that uses a non-systematic code, the original message is transformed into an encoded message carrying the same information and that has at least as many bits as the original message.

Good error control performance requires the scheme to be selected based on the characteristics of the communication channel. Common channel models include memoryless models where errors occur randomly and with a certain probability, and dynamic models where errors occur primarily in bursts. Consequently, error-detecting and correcting codes can be generally distinguished between random-error-detecting/correcting and burst-error-detecting/correcting. Some codes can also be suitable for a mixture of random errors and burst errors.

If the channel characteristics cannot be determined, or are highly variable, an error-detection scheme may be combined with a system for retransmissions of erroneous data. This is known as automatic repeat request (ARQ), and is most notably used in the Internet. An alternate approach for error control is hybrid automatic repeat request (HARQ), which is a combination of ARQ and error-correction coding.

Types of error correction[edit]

There are three major types of error correction.^[9]

Automatic repeat request[edit]

Automatic repeat request (ARQ) is an error control method for data transmission that makes use of error-detection codes, acknowledgment and/or negative acknowledgment messages, and timeouts to achieve reliable data transmission. An acknowledgment is a message sent by the receiver to indicate that it has correctly received a data frame.

Usually, when the transmitter does not receive the acknowledgment before the timeout occurs (i.e., within a reasonable amount of time after sending the data frame), it retransmits the frame until it is either correctly received or the error persists beyond a predetermined number of retransmissions.

Three types of ARQ protocols are Stop-and-wait ARQ, Go-Back-N ARQ, and Selective Repeat ARQ.

ARQ is appropriate if the communication channel has varying or unknown capacity, such as is the case on the Internet. However, ARQ requires the availability of a back channel, results in possibly increased latency due to retransmissions, and requires the maintenance of buffers and timers for retransmissions, which in the case of network congestion can put a strain on the server and overall network capacity.^[10]

For example, ARQ is used on shortwave radio data links in the form of ARQ-E, or combined with multiplexing as ARQ-M.

Forward error correction[edit]

Forward error correction (FEC) is a process of adding redundant data such as an error-correcting code (ECC) to a message so that it can be recovered by a receiver even when a number of errors (up to the capability of the code being used) are introduced, either during the process of transmission or on storage. Since the receiver does not have to ask the sender for retransmission of the data, a backchannel is not required in forward error correction. Error-correcting codes are used in lower-layer communication such as cellular network, high-speed fiber-optic communication and Wi-Fi,^[11][12] as well as for reliable storage in media such as flash memory, hard disk and RAM.^[13]

Error-correcting codes are usually distinguished between convolutional codes and block codes:

• Convolutional codes are processed on a bit-by-bit basis. They are particularly suitable for implementation in hardware, and the Viterbi decoder allows optimal decoding.
• Block codes are processed on a block-by-block basis. Early examples of block codes are repetition codes, Hamming codes and multidimensional parity-check codes. They were followed by a number of efficient codes, Reed–Solomon codes being the most notable due to their current widespread use. Turbo codes and low-density parity-check codes (LDPC) are relatively new constructions that can provide almost optimal efficiency.

Shannon’s theorem is an important theorem in forward error correction, and describes the maximum information rate at which reliable communication is possible over a channel that has a certain error probability or signal-to-noise ratio (SNR). This strict upper limit is expressed in terms of the channel capacity. More specifically, the theorem says that there exist codes such that with increasing encoding length the probability of error on a discrete memoryless channel can be made arbitrarily small, provided that the code rate is smaller than the channel capacity. The code rate is defined as the fraction k/n of k source symbols and n encoded symbols.

The actual maximum code rate allowed depends on the error-correcting code used, and may be lower. This is because Shannon’s proof was only of existential nature, and did not show how to construct codes which are both optimal and have efficient encoding and decoding algorithms.

Hybrid schemes[edit]

Hybrid ARQ is a combination of ARQ and forward error correction. There are two basic approaches:^[10]

• Messages are always transmitted with FEC parity data (and error-detection redundancy). A receiver decodes a message using the parity information, and requests retransmission using ARQ only if the parity data was not sufficient for successful decoding (identified through a failed integrity check).
• Messages are transmitted without parity data (only with error-detection information). If a receiver detects an error, it requests FEC information from the transmitter using ARQ, and uses it to reconstruct the original message.

The latter approach is particularly attractive on an erasure channel when using a rateless erasure code.

Error detection schemes[edit]

Error detection is most commonly realized using a suitable hash function (or specifically, a checksum, cyclic redundancy check or other algorithm). A hash function adds a fixed-length tag to a message, which enables receivers to verify the delivered message by recomputing the tag and comparing it with the one provided.

There exists a vast variety of different hash function designs. However, some are of particularly widespread use because of either their simplicity or their suitability for detecting certain kinds of errors (e.g., the cyclic redundancy check’s performance in detecting burst errors).

Minimum distance coding[edit]

A random-error-correcting code based on minimum distance coding can provide a strict guarantee on the number of detectable errors, but it may not protect against a preimage attack.

Repetition codes[edit]

A repetition code is a coding scheme that repeats the bits across a channel to achieve error-free communication. Given a stream of data to be transmitted, the data are divided into blocks of bits. Each block is transmitted some predetermined number of times. For example, to send the bit pattern «1011», the four-bit block can be repeated three times, thus producing «1011 1011 1011». If this twelve-bit pattern was received as «1010 1011 1011» – where the first block is unlike the other two – an error has occurred.

A repetition code is very inefficient, and can be susceptible to problems if the error occurs in exactly the same place for each group (e.g., «1010 1010 1010» in the previous example would be detected as correct). The advantage of repetition codes is that they are extremely simple, and are in fact used in some transmissions of numbers stations.^[14][15]

Parity bit[edit]

A parity bit is a bit that is added to a group of source bits to ensure that the number of set bits (i.e., bits with value 1) in the outcome is even or odd. It is a very simple scheme that can be used to detect single or any other odd number (i.e., three, five, etc.) of errors in the output. An even number of flipped bits will make the parity bit appear correct even though the data is erroneous.

Parity bits added to each «word» sent are called transverse redundancy checks, while those added at the end of a stream of «words» are called longitudinal redundancy checks. For example, if each of a series of m-bit «words» has a parity bit added, showing whether there were an odd or even number of ones in that word, any word with a single error in it will be detected. It will not be known where in the word the error is, however. If, in addition, after each stream of n words a parity sum is sent, each bit of which shows whether there were an odd or even number of ones at that bit-position sent in the most recent group, the exact position of the error can be determined and the error corrected. This method is only guaranteed to be effective, however, if there are no more than 1 error in every group of n words. With more error correction bits, more errors can be detected and in some cases corrected.

There are also other bit-grouping techniques.

Checksum[edit]

A checksum of a message is a modular arithmetic sum of message code words of a fixed word length (e.g., byte values). The sum may be negated by means of a ones’-complement operation prior to transmission to detect unintentional all-zero messages.

Checksum schemes include parity bits, check digits, and longitudinal redundancy checks. Some checksum schemes, such as the Damm algorithm, the Luhn algorithm, and the Verhoeff algorithm, are specifically designed to detect errors commonly introduced by humans in writing down or remembering identification numbers.

Cyclic redundancy check[edit]

A cyclic redundancy check (CRC) is a non-secure hash function designed to detect accidental changes to digital data in computer networks. It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a generator polynomial, which is used as the divisor in a polynomial long division over a finite field, taking the input data as the dividend. The remainder becomes the result.

A CRC has properties that make it well suited for detecting burst errors. CRCs are particularly easy to implement in hardware and are therefore commonly used in computer networks and storage devices such as hard disk drives.

The parity bit can be seen as a special-case 1-bit CRC.

Cryptographic hash function[edit]

The output of a cryptographic hash function, also known as a message digest, can provide strong assurances about data integrity, whether changes of the data are accidental (e.g., due to transmission errors) or maliciously introduced. Any modification to the data will likely be detected through a mismatching hash value. Furthermore, given some hash value, it is typically infeasible to find some input data (other than the one given) that will yield the same hash value. If an attacker can change not only the message but also the hash value, then a keyed hash or message authentication code (MAC) can be used for additional security. Without knowing the key, it is not possible for the attacker to easily or conveniently calculate the correct keyed hash value for a modified message.

Error correction code[edit]

Any error-correcting code can be used for error detection. A code with minimum Hamming distance, d, can detect up to d − 1 errors in a code word. Using minimum-distance-based error-correcting codes for error detection can be suitable if a strict limit on the minimum number of errors to be detected is desired.

Codes with minimum Hamming distance d = 2 are degenerate cases of error-correcting codes, and can be used to detect single errors. The parity bit is an example of a single-error-detecting code.

Applications[edit]

Applications that require low latency (such as telephone conversations) cannot use automatic repeat request (ARQ); they must use forward error correction (FEC). By the time an ARQ system discovers an error and re-transmits it, the re-sent data will arrive too late to be usable.

Applications where the transmitter immediately forgets the information as soon as it is sent (such as most television cameras) cannot use ARQ; they must use FEC because when an error occurs, the original data is no longer available.

Applications that use ARQ must have a return channel; applications having no return channel cannot use ARQ.

Applications that require extremely low error rates (such as digital money transfers) must use ARQ due to the possibility of uncorrectable errors with FEC.

Reliability and inspection engineering also make use of the theory of error-correcting codes.^[16]

Internet[edit]

In a typical TCP/IP stack, error control is performed at multiple levels:

• Each Ethernet frame uses CRC-32 error detection. Frames with detected errors are discarded by the receiver hardware.
• The IPv4 header contains a checksum protecting the contents of the header. Packets with incorrect checksums are dropped within the network or at the receiver.
• The checksum was omitted from the IPv6 header in order to minimize processing costs in network routing and because current link layer technology is assumed to provide sufficient error detection (see also RFC 3819).
• UDP has an optional checksum covering the payload and addressing information in the UDP and IP headers. Packets with incorrect checksums are discarded by the network stack. The checksum is optional under IPv4, and required under IPv6. When omitted, it is assumed the data-link layer provides the desired level of error protection.
• TCP provides a checksum for protecting the payload and addressing information in the TCP and IP headers. Packets with incorrect checksums are discarded by the network stack, and eventually get retransmitted using ARQ, either explicitly (such as through three-way handshake) or implicitly due to a timeout.

Deep-space telecommunications[edit]

The development of error-correction codes was tightly coupled with the history of deep-space missions due to the extreme dilution of signal power over interplanetary distances, and the limited power availability aboard space probes. Whereas early missions sent their data uncoded, starting in 1968, digital error correction was implemented in the form of (sub-optimally decoded) convolutional codes and Reed–Muller codes.^[17] The Reed–Muller code was well suited to the noise the spacecraft was subject to (approximately matching a bell curve), and was implemented for the Mariner spacecraft and used on missions between 1969 and 1977.

The Voyager 1 and Voyager 2 missions, which started in 1977, were designed to deliver color imaging and scientific information from Jupiter and Saturn.^[18] This resulted in increased coding requirements, and thus, the spacecraft were supported by (optimally Viterbi-decoded) convolutional codes that could be concatenated with an outer Golay (24,12,8) code. The Voyager 2 craft additionally supported an implementation of a Reed–Solomon code. The concatenated Reed–Solomon–Viterbi (RSV) code allowed for very powerful error correction, and enabled the spacecraft’s extended journey to Uranus and Neptune. After ECC system upgrades in 1989, both crafts used V2 RSV coding.

The Consultative Committee for Space Data Systems currently recommends usage of error correction codes with performance similar to the Voyager 2 RSV code as a minimum. Concatenated codes are increasingly falling out of favor with space missions, and are replaced by more powerful codes such as Turbo codes or LDPC codes.

The different kinds of deep space and orbital missions that are conducted suggest that trying to find a one-size-fits-all error correction system will be an ongoing problem. For missions close to Earth, the nature of the noise in the communication channel is different from that which a spacecraft on an interplanetary mission experiences. Additionally, as a spacecraft increases its distance from Earth, the problem of correcting for noise becomes more difficult.

Satellite broadcasting[edit]

The demand for satellite transponder bandwidth continues to grow, fueled by the desire to deliver television (including new channels and high-definition television) and IP data. Transponder availability and bandwidth constraints have limited this growth. Transponder capacity is determined by the selected modulation scheme and the proportion of capacity consumed by FEC.

Data storage[edit]

Error detection and correction codes are often used to improve the reliability of data storage media.^[19] A parity track capable of detecting single-bit errors was present on the first magnetic tape data storage in 1951. The optimal rectangular code used in group coded recording tapes not only detects but also corrects single-bit errors. Some file formats, particularly archive formats, include a checksum (most often CRC32) to detect corruption and truncation and can employ redundancy or parity files to recover portions of corrupted data. Reed-Solomon codes are used in compact discs to correct errors caused by scratches.

Modern hard drives use Reed–Solomon codes to detect and correct minor errors in sector reads, and to recover corrupted data from failing sectors and store that data in the spare sectors.^[20] RAID systems use a variety of error correction techniques to recover data when a hard drive completely fails. Filesystems such as ZFS or Btrfs, as well as some RAID implementations, support data scrubbing and resilvering, which allows bad blocks to be detected and (hopefully) recovered before they are used.^[21] The recovered data may be re-written to exactly the same physical location, to spare blocks elsewhere on the same piece of hardware, or the data may be rewritten onto replacement hardware.

Error-correcting memory[edit]

Dynamic random-access memory (DRAM) may provide stronger protection against soft errors by relying on error-correcting codes. Such error-correcting memory, known as ECC or EDAC-protected memory, is particularly desirable for mission-critical applications, such as scientific computing, financial, medical, etc. as well as extraterrestrial applications due to the increased radiation in space.

Error-correcting memory controllers traditionally use Hamming codes, although some use triple modular redundancy. Interleaving allows distributing the effect of a single cosmic ray potentially upsetting multiple physically neighboring bits across multiple words by associating neighboring bits to different words. As long as a single-event upset (SEU) does not exceed the error threshold (e.g., a single error) in any particular word between accesses, it can be corrected (e.g., by a single-bit error-correcting code), and the illusion of an error-free memory system may be maintained.^[22]

In addition to hardware providing features required for ECC memory to operate, operating systems usually contain related reporting facilities that are used to provide notifications when soft errors are transparently recovered. One example is the Linux kernel’s EDAC subsystem (previously known as Bluesmoke), which collects the data from error-checking-enabled components inside a computer system; besides collecting and reporting back the events related to ECC memory, it also supports other checksumming errors, including those detected on the PCI bus.^[23][24][25] A few systems^[specify] also support memory scrubbing to catch and correct errors early before they become unrecoverable.

See also[edit]

• Berger code
• Burst error-correcting code
• ECC memory, a type of computer data storage
• Link adaptation
• List of algorithms § Error detection and correction
• List of hash functions

References[edit]

Further reading[edit]

• Shu Lin; Daniel J. Costello, Jr. (1983). Error Control Coding: Fundamentals and Applications. Prentice Hall. ISBN 0-13-283796-X.
• SoftECC: A System for Software Memory Integrity Checking
• A Tunable, Software-based DRAM Error Detection and Correction Library for HPC
• Detection and Correction of Silent Data Corruption for Large-Scale High-Performance Computing

External links[edit]

• The on-line textbook: Information Theory, Inference, and Learning Algorithms, by David J.C. MacKay, contains chapters on elementary error-correcting 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, turbo codes, and fountain codes.
• ECC Page — implementations of popular ECC encoding and decoding routines

Introduction

Data storages and transmissions, regardless of the storage and transmission types, are not always error-free. There are always some non-zero probabilities that the data could be changed while it is being stored or transmitted.

For example, DNA is a perfect storage medium for information because DNA is highly stable, much more stable than any of the storage media used in the modern electronics, no matter whether it is the genetic material in vivo or the macromolecule in vitro. However, because of the natural radiations, DNA could always be damaged by radiations with some small probability. Therefore, the information the DNA stored could be altered or lost. In vivo, there are DNA repair mechanisms to fix the errors and damages brought by the radiation, usually based on the intact complementary information stored in the complementary DNA strand. DNA could also be altered while being transmitted, no matter whether it is being replicated by DNA polymerase in vivo or being synthesized from chemicals in vitro. In vivo, there are DNA error-correction mechanisms to fix the incorrect nucleotide being added by the DNA polymerase during DNA synthesis, usually also based on the intact complementary information stored in the complementary DNA strand.

An artificial data storage or transmission system would probably never be as delicate as the in vivo DNA genetic systems. Without some error-correction mechanisms for the artificial data storage and transmission system, the information could be lost forever. Early mathematicians and computer scientists, such as Richard Hamming, invented codes, that could automatically correct the errors introduced during data storage and transmission.

Hamming codes invented by Richard Hamming are a family of linear error-correcting codes and they have been widely used in applications such as error correction code memory (ECC memory). In this article, I would like to try pretending to be Richard Hamming and walk through how Hamming codes were invented mathematically.

Error Detection VS Error Correction

It should be noted that error detection and error correction are different. Usually error detection is much simpler and more efficient whereas error correction could be more complicated and less efficient.

In addition, error detection and error correction does not always work. There are always some small probabilities that error detection and error correction would run into situations that they were not designed to deal with.

Parity

Parity is one of most the simplest error detection codes. It adds a single bit that indicates whether the number of ones (bit-positions with values of one) in the preceding data was even or odd. If an odd number of bits is changed in transmission, the message will change parity and the error can be detected at this point; however, the bit that changed may have been the parity bit itself. The most common convention is that a parity value of one indicates that there is an odd number of ones in the data, and a parity value of zero indicates that there is an even number of ones. If the number of bits changed is even, the check bit will be valid and the error will not be detected.

The following is an example of attaching 1 parity bit to 7 bits of data, making the bit string to always have even number of 1s. Therefore, the XOR of the 8-bit data is always 0.

If the bit string has one single error, either in the data or at the parity bit position, the number of 1s in the bit string will not be even, and XOR will not be 0. However, if there are even number of errors in the bit string, the error could never be detected, as XOR would remain 0.

It is obvious that parity could not correct errors when it detects an error because it has no idea where the error happens. Whenever it detects an error, usually the receiver would have to request the sender to transmit the data again, which could be inconvenient and time-consuming.

Repetitions

Repetitions is one of most the simplest error correction codes. It repeats every data bit multiple times in order to ensure that it was sent correctly.

The following is an example of repeating every bit from 3 bits of data 3 times.

If the bit string has one single error, there would be inconsistency in the 3 bit repetition, and the single inconsistent bit would just be the error. The error could also be fixed by checking the values from the other 2 bits in the repetition. However, if there are two errors in the 3-bit repetition. For example, if the data including three repetitions 000 was incorrectly transmitted as 101, after correction, the data including three repetitions would become 111. The original data was 0, but it was incorrectly corrected to 1 after transmission and error correction.

The incorrect error correction could be mitigated by using larger number of repetitions. The more bit repetitions to vote, the more robust the error correction code to error rate. The drawback is it will reduce the error correction efficiency and is not favorable in a latency demanding data transmission system.

Hamming Codes

In mathematical terms, Hamming codes are a class of binary linear code. For each integer $r geq 2$, where $r$ is the number of error-correction bits, there is a code-word with block length $n = 2^r — 1$ and message length $k = 2^r — r — 1$.

For example, for $r = 3$, the Hamming code has $n = 7$ and $k = 4$. In the 7 bits of the Hamming code, 4 bits are the message we wanted to transmit, the rest 3 bits are error-correction bits which protects the message.

The rate of a block code, defined as the ratio between its message length and its block length, for Hamming codes is therefore

$$
begin{align}
R &= frac{k}{n} \
&= 1 — frac{r}{2^r — 1} \
end{align}
$$

As $r rightarrow infty$, $R rightarrow 1$. When $r$ is very large enough, almost all the bits in the Hamming code are messages.

The Hamming code algorithm would be “created” and derived in the following sections.

Hamming(7,4) Code

Let’s take a look at the core principles of Hamming(7,4) code, that is Hamming code when $r = 3$, first. We will try to come up ways of extending the Hamming(7,4) logic and principles to generic cases.

Hamming(7,4) code consists of 4 data bits, $d_1$, $d_2$, $d_3$, $d_4$, and 3 parity bits, $p_1$, $p_2$, $p_3$. As is shown in the following graphical depiction, the parity bit $p_1$ applies to the data bits $d_1$, $d_2$, and $d_4$, the parity bit $p_2$ applies to the data bits $d_1$, $d_3$, and $d_4$, the parity bit $p_3$ applies to the data bits $d_2$, $d_3$, and $d_4$.

When there is no error in the bits, none of the parities will break.

When there is a single error in the data bits, more than one parity will break. For example, say $d_1$ gets an error, the parities where $p_1$ and $p_2$ apply will break and the parity where $p_3$ applies remains. Then we could unambiguously determine $d_1$ gets an error. To correct the error, simply flip the value of $d_1$ as it’s binary.

When there is a single error in the parity bits, only one parity will break. For example, say $p_3$ gets an error, only the parity where $p_3$ applies will break and the parities where $p_1$ and $p_2$ apply remain.

We could see from Hamming(7,4) code that Hamming code is a natural extension from the parity used for error detection.

Create Generic Hamming Code From Scratch

Based on Hamming(7,4) code intuitions and principles, let’s pretend we were the genius Richard Hamming in 1950s and tried to come up with a generic Hamming code for different numbers of error-correction bits.

The number of parity bits $r$, and each parity bit is used only once in one parity. So there are $r$ parities, and the number of different parity states that Hamming code could represent is $2^r$. One parity state has to represent the state that the code has no error. Each of the rest parity states has to represent the state that one unique bit has an error. The number of the rest parity states is $2^r — 1$, therefore, $r$ parity bits could be used for error-correcting codes up to $2^r — 1$ bits. That’s why Hamming code has $r$ number of error-correction bits, the block length $n = 2^r — 1$ and message length $k = 2^r — r — 1$.

The next question is is it possible to and how to arrange $n = 2^r — 1$ bits into $r$ parities so that each single error could be represented by one unique parity state. The graphical depiction of Hamming(7,4) code perfectly tells us how to arrange Hamming code for $r = 3$.

But what’s the rationale behind this? How to arrange Hamming code for any $r$ that could potentially go very large?

The core principle is that no two bits from the $n = 2^r — 1$ bits will be applied by the same parity bit(s). The following mathematics is the guideline for arranging parity bits.

$$
begin{align}
n &= 2^r — 1 \
&= { {r}choose{0} } + { {r}choose{1} } + { {r}choose{2} } + cdots + { {r}choose{r} } — 1 \
&= { {r}choose{1} } + { {r}choose{2} } + cdots + { {r}choose{r} } \
end{align}
$$

Among the $n = 2^r — 1$ bits, ${r}choose{1}$ bits will be applied by only one parity bit, ${r}choose{2}$ bits will be applied by two parity bits, ${r}choose{3}$ bits will be applied by three parity bits, etc. More specifically, the ${r}choose{1}$ bits have to be the parity bits based on the principle that each parity bit is used only once in one parity.

Theoretically, we could randomly group the $k = n — {r choose 1} = 2^r — r — 1$ bits for the $r choose 2$, $r choose 3$, $cdots$, $r choose r$ groups. As long as the Hamming code decoder knows how the bits were grouped, the decoder could still decode the Hamming code to messages and correct the message based on $r$ parities.

Of course, random grouping does not seem to be optimal. So the next question related to optimization is can we group the bits in a way, rather than random, such that it is more friendly to Hamming code encoding and decoding and we could design efficient Hamming code encoding and decoding algorithms. More formally, given a sequence of $n = 2^r — 1$ bits, how to better assign $n = 2^r — 1$ bit (address) to $r$ parity bits and $k = 2^r — r — 1$ data bits and how to better assign $n = 2^r — 1$ bit (address) to $r$ parity groups?

A more motivating question to ask is, can we make those assignment such that by doing some algebra on the address of parity bits whose parities have been broken we would be able to find out the address of incorrect bits?

Maybe peeking into the Hamming(7,4) code bit assignments could give us some ideas. (I know we are cheating here since we are not really the genius Richard Hamming…)

We noticed that using one-based binary indexing, the addresses of parity bits $p_1$, $p_2$, and $p_3$ are 001, 010, 100, all of which are power of 2. What’s more interesting is, the address of the error bit could be computed from the address of parity bits whose parities have been broken using XOR. Going back to the examples in Hamming(7,4) code, suppose $d_1$ gets an error, the parities where $p_1$ and $p_2$ apply will break. The addresses of $d_1$, $p_1$, and $p_2$, are 011, 001, and 010, respectively. So, $011 = 001 text{XOR} 010$.

Knowing the address assignments of parity bits and data bits, it becomes straightforward to assign the bits to parity groups. To fill the following blanks, we will iterate through all the combinations of $p_1$, $p_2$, and $p_3$, excluding the null combination.

All the combinations of $p_1$, $p_2$, and $p_3$, excluding the null combination, are, ($p_1$), ($p_2$), ($p_3$), ($p_1$, $p_2$), ($p_1$, $p_3$), ($p_2$, $p_3$), ($p_1$, $p_2$, $p_3$). The index-XOR (address-XOR) for all the combinations are, 001, 010, 100, 011, 101, 110, 111, respectively. This tells us that bit 001 is guarded by parities where $p_1$ applies, bit 010 is guarded by parities where $p_2$ applies, …, bit 011 is guarded by parities where $p_1$ and $p_2$ applies, …, bit 111 is guarded by parities where $p_1$, $p_2$ and $p_3$ applies.

So the filled table will look like this.

More close examining reveals the following rules for parity bit assignments.

1. Parity bit $p_1$ covers all bit address which have the least significant bit set, 001, 011, 101, 111.
2. Parity bit $p_2$ covers all bit address which have the second least significant bit set, 010, 011, 110, 111.
3. Parity bit $p_3$ covers all bit address which have the third least significant bit set, 100, 101, 110, 111.

Strictly speaking, we would need a mathematical proof for deriving this rule. But we will just skip here.

The address assignment rules for parity bits and data bits and the parity bit assignment rules could be naturally extended to Hamming code for any $r geq 2$.

In addition, knowing exactly the parity group assignment is necessary for Hamming code encoding, but is useless for decoding, as decoding only needs to check the parity bits, compute the address of error bit using the parity bit address, and correct the error bit by flipping if necessary.

Generic Hamming Code Algorithm

Based on our creation of Hamming code from scratch, we could now state the generic Hamming code algorithm. Given a bit string,

1. Using one-based binary indexing for all the bits starting from 1, such as 00001, 00010, 00011, etc.
2. All bit positions that are powers of two (have a single 1 bit in the binary form of their position) are parity bits, such as 00001, 00010, 00100, 01000, etc.
3. All other bit positions, with two or more 1 bits in the binary form of their position, are data bits.
4. Each data bit is included in a unique set of 2 or more parity bits, as determined by the binary form of its bit position.
   • Parity bit $p_1$ covers all bit address which have the least significant bit set, 00001, 00011, 00101, 00111, etc.
   • Parity bit $p_2$ covers all bit address which have the second least significant bit set, 00010, 00011, 00110, 00111, etc.
   • Parity bit $p_3$ covers all bit address which have the third least significant bit set, 00100, 00101, 00110, 00111, etc.
   • …
   • Parity bit $p_i$ covers all bit address which have the i-th least significant bit set.

To encode the message, copy the message on to the data bit positions, and fill the parity bits by computing the parities based on the parity assignments.

To decode the message, check each of the parities. If any parity is broken, get the addresses of parity bits whose parity coverage are broken, and compute the error bit address using XOR for the addresses of broken parity bits that were just found. Once the error bit address was computed, flip the value of the error bit.

Hamming Code Dealing With Multiple Bit Errors

Let’s check one instance of Hamming(7,4) code 0110011 where $p_1 = 0$, $p_2 = 1$, $d_1 = 1$, $p_3 = 0$, $d_2 = 0$, $d_3 = 1$, $d_4 = 1$. Because of two bit errors were introduced, $p_3 = 1$, $d_2 = 1$, the Hamming(7,4) code becomes 0111111, as is shown below.

The parity where $p_1$ applies was broken suggesting that there is an error in the code. However, after fixing the code, the code will become 1111111 which is incorrect.

In general, in addition to Hamming(7,4) code, Hamming code can detect and correct a single error, but it cannot distinguish a double bit error of some codeword from a single bit error of a different codeword. Thus, some double-bit errors will be incorrectly decoded as if they were single bit errors and therefore go undetected, unless no correction is attempted.

Hamming Code With Additional Parity

To remedy the shortcoming of not being able to distinguish single bit error and double bit errors, Hamming codes can be extended by an extra parity bit. In this way, the decoder can detect and correct a single error and at the same time detect (but not correct) a double error.

This extra parity bit creates the parity with all the original Hamming code bits. When single bit error happens, the parity that the extra parity bit applies will break, in addition to the parity breaks from the original Hamming code. When double bit errors happen, the parity that the extra parity bit applies will remain intact, whereas the parity breaks from the original Hamming code remains unchanged.

The following is an example of Hamming(7,4) code with an extra parity bit, making it Hamming(8,4) code.

This extended Hamming code is popular in systems such as computer memory, as it will not try to make an correction mistake when double bit errors happen, which is already very rare.

Hamming Code (Binary) Linear Algebra

Hamming code encoding, error detection, error correction, and decoding processes could be represented using (binary) linear algebra.

Consider Hamming(7,4) code, the data bits could be represented using a column vector $mathbf{d} in mathbb{R}^{4}$, where

$$
begin{align}
mathbf{d} &=
begin{bmatrix}
d_1 \
d_2 \
d_3 \
d_4 \
end{bmatrix}
end{align}
$$

Encoding

To encode the data bits $mathbf{d}$, it is actually a (binary) linear transformation using the code generator matrix $mathbf{G}^{4 times 7}$, where

$$
begin{align}
mathbf{G}^{top} &=
begin{bmatrix}
1 & 1 & 0 & 1 \
1 & 0 & 1 & 1 \
1 & 0 & 0 & 0 \
0 & 1 & 1 & 1 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
end{bmatrix}
end{align}
$$

The Hamming(7,4) encoded bits $mathbf{x}$ would be

$$
begin{align}
mathbf{x} &= mathbf{G}^{top} mathbf{d} \
&=
begin{bmatrix}
1 & 1 & 0 & 1 \
1 & 0 & 1 & 1 \
1 & 0 & 0 & 0 \
0 & 1 & 1 & 1 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
end{bmatrix}
begin{bmatrix}
d_1 \
d_2 \
d_3 \
d_4 \
end{bmatrix} \
&=
begin{bmatrix}
d_1 + d_2 + d_4 \
d_1 + d_3 + d_4 \
d_1 \
d_2 + d_3 + d_4 \
d_2 \
d_3 \
d_4 \
end{bmatrix} \
&=
begin{bmatrix}
d_1 + d_2 + d_4 \
d_1 + d_3 + d_4 \
d_1 \
d_2 + d_3 + d_4 \
d_2 \
d_3 \
d_4 \
end{bmatrix}
mod 2 \
&=
begin{bmatrix}
(d_1 + d_2 + d_4) mod 2 \
(d_1 + d_3 + d_4) mod 2 \
d_1 \
(d_2 + d_3 + d_4) mod 2 \
d_2 \
d_3 \
d_4 \
end{bmatrix} \
&=
begin{bmatrix}
p_1 \
p_2 \
d_1 \
p_3 \
d_2 \
d_3 \
d_4 \
end{bmatrix} \
end{align}
$$

The modulo is to compute the parity bits for even parity.

Also notice that the code generator matrix $mathbf{G}$ could be easily derived from the bit sequence and parity table.

Error Detection

Suppose the actual Hamming(7,4) encoded bits received is $mathbf{x}^{prime}$, and $mathbf{x}^{prime}$ may or may not equal to $mathbf{x}$.

The error detection is just parity checking by applying the parity checking matrix $mathbf{H} in mathbb{R}^{3 times 7}$ on $mathbf{x}^{prime}$, where

$$
begin{align}
mathbf{H} &=
begin{bmatrix}
1 & 0 & 1 & 0 & 1 & 0 & 1 \
0 & 1 & 1 & 0 & 0 & 1 & 1 \
0 & 0 & 0 & 1 & 1 & 1 & 1 \
end{bmatrix}
end{align}
$$

$$
begin{align}
mathbf{z} = mathbf{H} mathbf{x}^{prime}
end{align}
$$

If $mathbf{x} = mathbf{x}^{prime}$, we must have $mathbf{z} = mathbf{0}$.

$$
begin{align}
mathbf{z} &= mathbf{H} mathbf{x}^{prime} \
&= mathbf{H} mathbf{x} \
&=
begin{bmatrix}
1 & 0 & 1 & 0 & 1 & 0 & 1 \
0 & 1 & 1 & 0 & 0 & 1 & 1 \
0 & 0 & 0 & 1 & 1 & 1 & 1 \
end{bmatrix}
begin{bmatrix}
p_1 \
p_2 \
d_1 \
p_3 \
d_2 \
d_3 \
d_4 \
end{bmatrix} \
&=
begin{bmatrix}
p_1 + d_1 + d_2 + d_3 \
p_2 + d_1 + d_3 + d_4 \
p_3 + d_d + d_3 + d_4 \
end{bmatrix} \
&=
begin{bmatrix}
p_1 + d_1 + d_2 + d_3 \
p_2 + d_1 + d_3 + d_4 \
p_3 + d_d + d_3 + d_4 \
end{bmatrix} mod 2 \
&=
begin{bmatrix}
0 \
0 \
0 \
end{bmatrix} \
&= mathbf{0} \
end{align}
$$

Also notice that the parity checking matrix $mathbf{H}$ is just the bit sequence and parity table.

A more interesting property of $mathbf{H}$ is that each column in $mathbf{H}$ is actually binary index sequence, 1, 2, 3, etc.

$$
begin{align}
mathbf{H} &=
begin{bmatrix}
1 & 0 & 1 & 0 & 1 & 0 & 1 \
0 & 1 & 1 & 0 & 0 & 1 & 1 \
0 & 0 & 0 & 1 & 1 & 1 & 1 \
end{bmatrix} \
&=
begin{bmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7 \
end{bmatrix}_{10}
end{align}
$$

This property will be very useful for error correction.

Error Correction

Suppose there is a single bit error at bit position $i$ (one-based), the actual Hamming encoded code received would be

$$
mathbf{x}^{prime} = mathbf{x} + mathbf{e}_{i}
$$

Parity checking would just yield the bit error position in binary.

$$
begin{align}
mathbf{z} &= mathbf{H} mathbf{x}^{prime} \
&= mathbf{H} left( mathbf{x} + mathbf{e}_{i} right) \
&= mathbf{H} mathbf{x} + mathbf{H} mathbf{e}_{i} \
&= mathbf{0} + mathbf{H} mathbf{e}_{i} \
&= mathbf{H} mathbf{e}_{i} \
&=
begin{bmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7 \
end{bmatrix}_{10}
mathbf{e}_{i} \
&= i \
&= (i)_{2} \
end{align}
$$

The resulting value would be non-zero if there is a bit error.

To fix the bit error, simply flip the bit value at address $mathbf{z}$.

Decoding

Decoding is simple. Suppose the corrected Hamming code is $mathbf{d}_{r}$, and the decoding matrix $mathbf{R} in mathbb{R}^{4 times 7}$, where

$$
begin{align}
mathbf{R} &=
begin{bmatrix}
0 & 0 & 1 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 1 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 1 \
end{bmatrix}
end{align}
$$

Also notice that the code generator matrix $mathbf{R}$ could be easily derived from the bit sequence and parity table.

Perfect Code

Minimum Distance

The distance or minimum (Hamming) distance $d$ of a block code is the minimum number of positions in which any two distinct codewords differ.

Because each message bit is covered by at least two parities, changing one message bit will result in changing at least three bits in codeword. Changing two message bits will result in changing at least three bits in codeword as well. So the distance of Hamming code $d = 3$.

In fact, a code with distance $d$ allows the receiver to detect up to $d-1$ transmission errors since changing $d-1$ positions of a codeword can never accidentally yield another codeword. Furthermore, if no more than $(d-1)/2$ transmission errors occur, the receiver can uniquely decode the received word to a codeword. This is because every received word has at most one codeword at distance $(d-1)/2$. If more than $(d-1)/2$ transmission errors occur, the receiver cannot uniquely decode the received word in general as there might be several possible codewords.

To understand this more intuitively, suppose we have certain number of people standing on a playground, the distance can only be integer, and the minimum distance between any two people is $d$. Given a circle of a radius of $(d-1)/2$ around each people, we know the closest person to all the points in the circle is the person in the center of the circle. If the radius of circles becomes $d — 1$, there are some chances that the closest person to all the points in the circle is not the person in the center of the circle, but the choices of the closest person is very limited. If the radius of circles becomes even larger, it is almost impossible to have the closest person to be the one in the center of the circle.

Hamming Bound

With the understanding of minimum distance from Hamming code, we could further derive Hamming bound, an important limitation on the efficiency with which any error-correcting code can utilize the space in which its code words are embedded. A code that attains the Hamming bound is said to be a perfect code.

An alphabet set $mathcal{A}_{q}$ is a set of symbols with $q$ elements and $lvert mathcal{A}_{q} rvert = q$. The set of strings of length $n$ on the alphabet set $mathcal{A}_{q}$ are denoted $mathcal{A}_{q}^{n}$ and $lvert mathcal{A}_{q}^{n} rvert = q^n$. A $q$-ary block code of length $n$ is a subset of the strings of $mathcal{A}_{q}^{n}$.

Let $A_{q}(n,d)$ denote the maximum possible size of a $q$-ary block code $C$ of length $n$ and minimum Hamming distance $d$ between elements of the block code (necessarily positive for $q^{n}>1$).

Then, the Hamming bound is:

$$
A_{q}(n,d) leq frac{q^n}{sum_{k=0}^{t} {n choose k} (q-1)^k }
$$

where

$$
t = leftlfloor frac{d-1}{2} rightrfloor
$$

Proof

Similar to what we have discussed in the minimum distance section, for each codeword $c$ in $q$-ary block code $C$, consider a ball of fixed radius of $t$ around $c$, where $t$ is just the minimum distance. The number of words $m$ that could deviate at most $t$ components from the center of the sphere, which is a codeword, could be computed using combinations.

$$
m = sum_{k=0}^{t} {n choose k} (q-1)^k
$$

Because $A_{q}(n,d)$ is the maximum number of codewords in $C$, we must have

$$
A_{q}(n,d) times m leq lvert mathcal{A}_{q}^{n} rvert
$$

Therefore,

$$
A_{q}(n,d) leq frac{q^n}{sum_{k=0}^{t} {n choose k} (q-1)^k }
$$

This concludes the proof. $square$

Hamming Codes

A code that attains the Hamming bound is said to be a perfect code. Perfect code achieves the highest possible rate for codes with their block length and minimum distance.

Hamming codes are perfect codes. Let’s verify it.

Proof

For Hamming codes, as we have derived previously, $q$ = 2 because Hamming codes are binary codes, code length $n = 2^r — 1$, message length $k = 2^r — r — 1$, and minimum distance $t = lfloor (d — 1) / 2 rfloor = 1$.

$$
begin{align}
A_{q}(n,d) &= q^{k} \
&= 2^{2^r — r — 1} \
end{align}
$$

$$
begin{align}
frac{q^n}{sum_{i=0}^{t} {n choose i} (q-1)^i }
&= frac{2^n}{sum_{i=0}^{1} {n choose i} 1^i } \
&= frac{2^n}{ {n choose 0} + {n choose 1} } \
&= frac{2^n}{ 1 + n } \
&= frac{2^{2^r — 1}}{ 1 + 2^r — 1 } \
&= frac{2^{2^r — 1}}{ 2^r } \
&= 2^{2^r — r — 1} \
end{align}
$$

Therefore, for Hamming codes,

$$
A_{q}(n,d) = frac{q^n}{sum_{k=0}^{t} {n choose k} (q-1)^k }
$$

Hamming code is perfect code.

This concludes the proof. $square$

As I mentioned, random parity grouping would also work in Hamming code for error correction but it would just be less efficient. The way Richard Hamming was assigning the bits to parity groups was the only part that we “cheated” in this article. There must be some mathematical motivations behind this, which I did not get time to research and explore.

That’s why Richard Hamming was a famous mathematician but I am not

References

• Hamming Code
• Hamming Distance
• Hamming Bound

In mathematics, computer science, telecommunication, and information theory, error detection and correction has great practical importance in maintaining data (information) integrity across noisy channels and less-than-reliable storage media.

General definitions of terms

Definitions of Error detection and error correction:
* Error detection is the ability to detect the presence of errors caused by noise or other impairments during transmission from the transmitter to the receiver.
* Error correction is the additional ability to reconstruct the original, error-free data.

There are two basic ways to design the channel code and protocol for an error correcting system:
* Automatic repeat-request (ARQ): The transmitter sends the data and also an error detection code, which the receiver uses to check for errors, and requests retransmission of erroneous data. In many cases, the request is implicit; the receiver sends an acknowledgement (ACK) of correctly received data, and the transmitter re-sends anything not acknowledged within a reasonable period of time.
* Forward error correction (FEC): The transmitter encodes the data with an error-correcting code (ECC) and sends the coded message. The receiver never sends any messages back to the transmitter. The receiver decodes what it receives into the «most likely» data. The codes are designed so that it would take an «unreasonable» amount of noise to trick the receiver into misinterpreting the data.

It is possible to combine the two, so that minor errors are corrected without retransmission, and major errors are detected and a retransmission requested. The combination is called hybrid automatic repeat-request.

Error detection schemes

In telecommunication, a «redundancy check» is extra data added to a message for the purposes of error detection.

Several schemes exist to achieve error detection, and are generally quite simple. All error detection codes (which include all error-detection-and-correction codes) transmit more bits than were in the original data. Most codes are «systematic»: the transmitter sends a fixed number of original data bits, followed by fixed number of «check bits» (usually referred to as redundancy in the literature) which are derived from the data bits by some deterministic algorithm. The receiver applies the same algorithm to the received data bits and compares its output to the received check bits; if the values do not match, an error has occurred at some point during the transmission. In a system that uses a «non-systematic» code, such as some raptor codes, data bits are transformed into at least as many code bits, and the transmitter sends only the code bits.

Repetition schemes

Variations on this theme exist. Given a stream of data that is to be sent, the data is broken up into blocks of bits, and in sending, each block is sent some predetermined number of times. For example, if we want to send «1011», we may repeat this block three times each.

Suppose we send «1011 1011 1011», and this is received as «1010 1011 1011». As one group is not the same as the other two, we can determine that an error has occurred. This scheme is not very efficient, and can be susceptible to problems if the error occurs in exactly the same place for each group (e.g. «1010 1010 1010» in the example above will be detected as correct in this scheme).

The scheme however is extremely simple, and is in fact used in some transmissions of numbers stations.Fact|date=July 2008

Parity schemes

:»Main article»: Parity bitA «parity bit» is an «error detection» mechanism that can only detect an odd number of errors.

The stream of data is broken up into blocks of bits, and the number of 1 bits is counted. Then, a «parity bit» is set (or cleared) if the number of one bits is odd (or even). (This scheme is called even parity; odd parity can also be used.) If the tested blocks overlap, then the parity bits can be used to isolate the error, and even correct it if the error affects a single bit: this is the principle behind the Hamming code.

There is a limitation to parity schemes. A parity bit is only guaranteed to detect an odd number of bit errors (one, three, five, and so on). If an even number of bits (two, four, six and so on) are flipped, the parity bit appears to be correct, even though the data is corrupt.

Checksum

: «Main article»: ChecksumA checksum of a message is an arithmetic sum of message code words of a certain word length, for example byte values, and their carry value. The sum is negated by means of ones-complement, and stored or transferred as an extra code word extending the message.

On the receiver side, a new checksum may be calculated, from the extended message. If the new checksum is not 0, error is detected.

Checksum schemes include parity bits, check digits, and longitudinal redundancy check.

Cyclic redundancy checks

: «Main article»: Cyclic redundancy checkMore complex error detection (and correction) methods make use of the properties of finite fields and polynomials over such fields.

The cyclic redundancy check considers a block of data as the coefficients to a polynomial and then divides by a fixed, predetermined polynomial. The coefficients of the result of the division is taken as the redundant data bits, the CRC.

On reception, one can recompute the CRC from the payload bits and compare this with the CRC that was received. A mismatch indicates that an error occurred.

Hamming distance based checks

If we want to detect «d» bit errors in an «n» bit word we can map every «n» bit word into a bigger «n+d+1» bit word so that the minimum Hamming distance between each valid mapping is «d+1». This way, if one receives a «n+d+1» word that doesn’t match any word in the mapping (with a Hamming distance «x» <= «d+1» from any word in the mapping) it can successfully detect it as an errored word. Even more, «d» or fewer errors will never transform a valid word into another, because the Hamming distance between each valid word is at least «d+1», and such errors only lead to invalid words that are detected correctly.Given a stream of «m*n» bits, we can detect «x <= d» bit errors successfully using the above method on every «n» bit word. In fact, we can detect a maximum of «m*d» errors if every «n» word is transmitted with maximum «d» errors.

Hash function

Any hash function can be used as a redundancy check.

Horizontal and vertical redundancy check

Other types of redundancy check include horizontal redundancy check, vertical redundancy check and RAID#Non-standard_levels «double», «dual» or «diagonal» parity (used in Raid-DP).

Polarity schemes

One less commonly used form of error correction and detection is transmitting a polarity reversed bitstream simultaneously with the bitstream it is meant to correct. This scheme is very weak at detecting bit errors, and marginally useful for byte or word error detection and correction. However, at the physical layer in the OSI model, this scheme can aid in error correction and detection.

Polarity symbol reversal is (probably) the simplest form of Turbo code, but technically not a Turbo code at all.
* Turbo codes DO NOT work at the bit level.
* Turbo codes typically work at the character or symbol level depending on their placement in the OSI model.
* Character here refers to Baudot, ASCII-7, the 8-bit byte or the 16-bit word.

Original transmitted symbol «1011»
* transmit 1011 on carrier wave 1 (CW1)
* transmit 0100 on carrier wave 2 (CW2)

Receiver end
* do bits polarities of (CW1) <> (CW2)?
* if CW1 = CW2, signal «bit error» (triggers more complex ECC)

This polarity reversal scheme works fairly well at low data rates (below 300 baud) with very redundant data like telemetry data.Specify|date=February 2007

Error correction

Automatic repeat request

Automatic Repeat-reQuest (ARQ) is an error control method for data transmission which makes use of error detection codes, acknowledgment and/or negative acknowledgement messages and timeouts to achieve reliable data transmission. An acknowledgment is a message sent by the receiver to the transmitter to indicate that it has correctly received a data frame.

Usually, when the transmitter does not receive the acknowledgment before the timeout occurs (i.e. within a reasonable amount of time after sending the data frame), it retransmits the frame until it is either correctly received or the error persists beyond a predetermined number of retransmissions.

A few types of ARQ protocols are Stop-and-wait ARQ, Go-Back-N ARQ and Selective Repeat ARQ.

Hybrid ARQ is a combination of ARQ and forward error correction.

Error-correcting code

An error-correcting code (ECC) or forward error correction (FEC) code is a code in which each data signal conforms to specific rules of construction so that departures from this construction in the received signal can generally be automatically detected and corrected. It is used in computer data storage, for example in dynamic RAM, and in data transmission.

The basic idea is for the transmitter to apply one or more of the above error detecting codes;then the receiver uses those codes to narrow down exactly where in the message the error (if any) is.If there is a single bit error in transmission, the decoder uses those error detecting codes to narrow down the error to a single bit (1 or 0), then fix that error by flipping that bit. (Later we will discuss ways of dealing with more than one error per message.)

* repetition schemes — if the transmitter repeats each data bit at least 3 different times (triple modular redundancy), the receiver can correct any single-bit error by taking the majority vote of the received data bits.
* parity schemes — if the transmitter sends parity bits covering various overlapping groups of the data bits, a single-bit error will cause a parity error in every group that covers that erroneous bit. The receiver can correct any single-bit error by flipping the one bit covered by every group that indicates an error, but not covered by any group that checks out good. There are a wide variety of parity-based codes, differing in exactly how groups of data bits are chosen. We will discuss them later.
* cyclic redundancy checks — when a transmitter adds a CRC code to a message, any single-bit error will cause the received CRC to differ from the receiver-calculated CRC. If the message is short enough, the receiver can figure out exactly which bit was flipped, and correct it — Header Error Correction.
* Hamming distance based checks — Since it takes many bit errors to convert one valid Hamming code word to any other valid Hamming code word, the receiver can correct any single-bit error in a word by finding the «closest» valid Hamming code, the one code word that has only 1 bit different from the received word.

Some codes can correct a certain number of bit errors and only detect further numbers of bit errors. Codes which can correct one error are termed single error correcting (SEC), and those which detect two are termed double error detecting (DED). Hamming codes can correct single-bit errors and detect double-bit errors (SEC-DED) — more sophisticated codes correct and detect even more errors.

An error-correcting code which corrects all errors of up to «n» bits correctly is also an error-detecting code which can detect at least all errors of up to 2″n» bits.

Two main categories are convolutional codes and block codes. Examples of the latter are Hamming code, BCH code, Reed-Solomon code, Reed-Muller code, Binary Golay code, and low-density parity-check codes.

Shannon’s theorem is an important theorem in error correction which describes the maximum attainable efficiency of an error-correcting scheme versus the levels of noise interference expected. In general, these methods put redundant information into the data stream following certain algebraic or geometric relations so that the decoded stream, if damaged in transmission, can be corrected. The effectiveness of the coding scheme is measured in terms of code rate, which is the code length divided by the useful information, and the Coding gain, which is the difference of the SNR levels of the uncoded and coded systems required to reach the same BER levels.

Error-correcting memory

Because soft errors are extremely common in the DRAM of computers used in satellites and space probes, such memory is structured as ECC memory (also called «EDAC protected memory»).Typically every bit of memory is refreshed at least 15 times per second.During this memory refresh, the memory controller reads each word of memory and writes the (corrected) word back. Fact|date=April 2007Such memory controllers traditionally use a Hamming code, although some use triple modular redundancy.Even though a single cosmic ray can upset many physically neighboring bits in a DRAM, such memory systems are designed so that neighboring bits belong to different words, so such single event upsets (SEUs) cause only a single error in any particular word, and so can be corrected by a single-bit error correcting code.As long as no more than a single bit in any particular word is hit by an error between refreshes, such a memory system presents the illusion of an error-free memory. [http://www.apmcsta.org/doc/Chen%20Zhenyu.doc ]

Applications

Applications that require low latency (such as telephone conversations) cannot use Automatic Repeat reQuest (ARQ); they must use Forward Error Correction (FEC). By the time an ARQ system discovers an error and re-transmits it, the re-sent data will arrive too late to be any good.

Applications where the transmitter immediately forgets the information as soon as it is sent (such as most television cameras) cannot use ARQ; they must use FEC because when an error occurs, the original data is no longer available. (This is also why FEC is used in data storage systems such as RAID and distributed data store).

Applications that use ARQ must have a return channel. Applications that have no return channel cannot use ARQ.

Applications that require extremely low error rates (such as digital money transfers) must use ARQ.

The Internet

In a typical TCP/IP stack, error detection is performed at multiple levels:
* Each Ethernet frame carries a CRC-32 checksum. The receiver discards frames if their checksums do not match.
* The IPv4 header contains a header checksum of the contents of the header (excluding the checksum field). Packets with checksums that don’t match are discarded.
* The checksum was omitted from the IPv6 header, because most current link layer protocols have error detection.
* UDP has an optional checksum. Packets with wrong checksums are discarded.
* TCP has a checksum of the payload, TCP header (excluding the checksum field) and source- and destination addresses of the IP header. Packets found to have incorrect checksums are discarded and eventually get retransmitted when the sender receives a triple-ack or a timeout occurs.

Deep-space telecommunications

NASA has used many different error correcting codes. For missions between 1969 and 1977 the Mariner spacecraft used a Reed-Muller code. The noise these spacecraft were subject to was well approximated by a «bell-curve» (normal distribution), so the Reed-Muller codes were well suited to the situation.

The Voyager 1 & Voyager 2 spacecraft transmitted color pictures of Jupiter and Saturn in 1979 and 1980.
* Color image transmission required 3 times the amount of data, so the Golay (24,12,8) code was used.Fact|date=December 2007
* This Golay code is only 3-error correcting, but it could be transmitted at a much higher data rate.
* Voyager 2 went on to Uranus and Neptune and the code was switched to a concatenated Reed-Solomon code-Convolutional code for its substantially more powerful error correcting capabilities.
* Current DSN error correction is done with dedicated hardware.
* For some NASA deep space craft such as those in the Voyager program, Cassini-Huygens (Saturn), New Horizons (Pluto) and Deep Space 1 — the use of hardware ECC may not be feasible for the full duration of the mission.

The different kinds of deep space and orbital missions that are conducted suggest that trying to find a «one size fits all» error correction system will be an ongoing problem for some time to come.
* For missions close to the earth the nature of the «noise» is different from that on a spacecraft headed towards the outer planets.
* In particular, if a transmitter on a spacecraft far from earth is operating at a low power, the problem of correcting for noise gets larger with distance from the earth.

Satellite broadcasting (DVB)

The demand for satellite transponder bandwidth continues to grow, fueled by the desire to deliver television (including new channels and High Definition TV) and IP data. Transponder availability and bandwidth constraints have limited this growth, because transponder capacity is determined by the selected modulation scheme and Forward error correction (FEC) rate.

Overview
* QPSK coupled with traditional Reed Solomon and Viterbi codes have been used for nearly 20 years for the delivery of digital satellite TV.
* Higher order modulation schemes such as 8PSK, 16QAM and 32QAM have enabled the satellite industry to increase transponder efficiency by several orders of magnitude.
* This increase in the information rate in a transponder comes at the expense of an increase in the carrier power to meet the threshold requirement for existing antennas.
* Tests conducted using the latest chipsets demonstrate that the performance achieved by using Turbo Codes may be even lower than the 0.8 dB figure assumed in early designs.

Data storage

Error detection and correction codes are often used to improve the reliability of data storage media.

A «parity track» was present on the first magnetic tape data storage in 1951. The «Optimal Rectangular Code» used in group code recording tapes not only detects but also corrects single-bit errors.

Some file formats, particularly archive formats, include a checksum (most often CRC32) to detect corruption and truncation and can employ redundancy and/or parity files to recover portions of corrupted data.

Reed Solomon codes are used in compact discs to correct errors caused by scratches.

Modern hard drives use CRC codes to detect and Reed-Solomon codes to correct minor errors in sector reads, and to recover data from sectors that have «gone bad» and store that data in the spare sectors. [ [http://www.myharddrivedied.com/presentations_whitepaper.html My Hard Drive Died | Scott A. Moulton ] ]

RAID systems use a variety of error correction techniques, to correct errors when a hard drive completely fails.

The Vandermonde matrix is used in some RAID techniques.

Information theory and error detection and correction

Information theory tells us that whatever the probability of error in transmission or storage, it is possible to construct error-correcting codes in which the likelihood of failure is arbitrarily low, although this requires adding increasing amounts of redundant data to the original, which might not be practical when the error probability is very high. Shannon’s theorem sets an upper bound to the error correction rate that can be achieved (and thus the level of noise that can be tolerated) using a fixed amount of redundancy, but does not tell us how to construct such an optimal encoder.

Error-correcting codes can be divided into block codes and convolutional codes. Other block error-correcting codes, such as Reed-Solomon codes, transform a chunk of bits into a (longer) chunk of bits in such a way that errors up to some threshold in each block can be detected and corrected.

However, in practice errors often occur in bursts rather than at random. This is often compensated for by shuffling (interleaving) the bits in the message after coding. Then any burst of bit-errors is broken up into a set of scattered single-bit errors when the bits of the message are unshuffled (de-interleaved) before being decoded.

List of error-correction, error-detection methods

This list contains methods of error correction (Reed-Solomon, for example is a method) and practical techniques for error correction (like the Check digit, a practical method).
* Berger code
* Chipkill, an application of ECC techniques to volatile system memory.
* Constant-weight code
* Convolutional codes are usually decoded with iterative Viterbi decoding techniques
* Differential space–time codes, related to space–time block codes.
* Dual modular redundancy, subset of N-modular redundancy, related to triple modular redundancy
* Erasure codes are a superset of Fountain codes
* Forward error correction
* Group code
* Golay code, the Binary Golay codes are the most commonly used Golay codes
* Goppa code that is used to create the McEliece cryptosystem
* Hadamard code
* Hagelbarger code
* Hamming code
* Lexicographic code
* Longitudinal redundancy check
* Low-density parity-check code
* LT codes are near optimal rateless erasure correcting codes.
* m of n codes
* Online codes are an example of rateless erasure codes.
* Parity bit
* Raptor codes are high speed (near real time) fountain codes.
* Reed-Solomon error correction
* Reed-Muller code
* Repeat-accumulate code
* Sparse graph code
* Space–time code
* Space–time trellis code
* Tornado codes are optimal Fountain codes
* Triple modular redundancy
* Turbo code
* Viterbi algorithm
* Walsh code used in cellular telephony for its high noise immunity, not just its ECC capabilities

Practical uses of Error Correction methods
* Concatenated error correction codes, the Compact Disc and Voyager Program spacecraft use concatenated error correction technologies
* Check digit, commonly used on UPC barcodes
* Luhn algorithm, the most commonly used base 10 checksum that can perform limited error detection but not error correction
* Luhn mod N algorithm, the above algorithm but implementable in a non base 10 form
* Verhoeff algorithm, a modular based form not related to the Luhn algorithms that can detect most forms of transposition errors in financial cryptographic applications

See also

* Repetition schemes:* Triple modular redundancy
* Error correction standardization:* Federal Standard 1037C:* MIL-STD-188
* Errors and residuals in statistics
* Research Conferences on Error Correction:* International Symposium On Turbo Codes, http://www-turbo.enst-bretagne.fr/

References

External links

* [http://www.inference.phy.cam.ac.uk/mackay/itila/ The on-line textbook: Information Theory, Inference, and Learning Algorithms] , by David MacKay, contains chapters on elementary error-correcting 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, turbo codes, and fountain codes.
* Article: [http://cr.yp.to/hardware/ecc.html Memory errors and SECDED]
* [http://ipsit.bu.edu/comp.html Compute parameters of linear codes] — an on-line interface for generating and computing parameters (e.g. minimum distance, covering radius) of linear error-correcting codes.
* [http://www.eccpage.com/ ECC Page]
* [http://gaussianwaves.blogspot.com Digital communication blog ]