# Error detection and correction

In and coding theory with applications in computer science and , error detection and correction or error control are techniques that enable reliable delivery of over unreliable . Many communication channels are subject to ,<!–really need an article on –> 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

The general definitions of the terms are as follows:

• 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

The modern development of in 1947 is due to . A description of Hamming’s code appeared in ‘s A Mathematical Theory of Communication and was quickly generalized by .

## Introduction

The general idea for achieving error detection and correction is to add some (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 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 . 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 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 include memory-less models where errors occur randomly and with a certain probability, and dynamic models where errors occur primarily in . 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 cannot be determined, or is 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.

## Implementation

Error correction may generally be realized in two different ways:

• Automatic repeat request (ARQ) (sometimes also referred to as backward error correction): This is an error control technique whereby an error detection scheme is combined with requests for retransmission of erroneous data. Every block of data received is checked using the error detection code used, and if the check fails, retransmission of the data is requested – this may be done repeatedly, until the data can be verified.
• Forward error correction (FEC): The sender encodes the data using an error-correcting code (ECC) prior to transmission. The additional information (redundancy) added by the code is used by the receiver to recover the original data. In general, the reconstructed data is what is deemed the “most likely” original data.

ARQ and FEC may be combined, such that minor errors are corrected without retransmission, and major errors are corrected via a request for retransmission: this is called (HARQ).

## Error detection schemes

Error detection is most commonly realized using a suitable hash function (or checksum 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 ).

A random-error-correcting code based on coding can provide a strict guarantee on the number of detectable errors, but it may not protect against a . A repetition code, described in the section below, is a special case of error-correcting code: although rather inefficient, a repetition code is suitable in some applications of error correction and detection due to its simplicity.

### Repetition codes

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”. However, if this twelve-bit pattern was received as “1010 1011 1011” – where the first block is unlike the other two – it can be determined that 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 .

### Parity bits

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.

Extensions and variations on the parity bit mechanism are , , and “double,” “dual,” or “diagonal” parity (used in ).

### Checksums

A checksum of a message is a sum of message code words of a fixed word length (e.g., byte values). The sum may be negated by means of a operation prior to transmission to detect errors resulting in 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 checks (CRCs)

A cyclic redundancy check (CRC) is a non-secure hash function designed to detect accidental changes to digital data in computer networks; as a result, it is not suitable for detecting maliciously introduced errors. It is characterized by specification of what is called a generator polynomial, which is used as the in a over a , taking the input data as the , such that the becomes the result.

A cyclic code has favorable properties that make it well suited for detecting . CRCs are particularly easy to implement in hardware, and are therefore commonly used in digital and storage devices such as .

Even parity is a special case of a cyclic redundancy check, where the single-bit CRC is generated by the divisor x + 1.

### Cryptographic hash functions

The output of a cryptographic hash function, also known as a message digest, can provide strong assurances about , 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 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 infeasible for the attacker to calculate the correct keyed hash value for a modified message.

### Error-correcting codes

Any error-correcting code can be used for error detection. A code with minimum , 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.

## Error correction

### Automatic repeat request (ARQ)

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 to achieve reliable data transmission. An acknowledgment is a message sent by the receiver to indicate that it has correctly received a .

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 , such as is the case on the Internet. However, ARQ requires the availability of a , results in possibly increased due to retransmissions, and requires the maintenance of buffers and timers for retransmissions, which in the case of can put a strain on the server and overall network capacity.

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

### Error-correcting code

An (ECC) or forward error correction (FEC) code is a process of adding data, or parity data, to a message, such that it can be recovered by a receiver even when a number of errors (up to the capability of the code being used) were 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 is not required in forward error correction, and it is therefore suitable for such as . Error-correcting codes are frequently used in communication, as well as for reliable storage in media such as , , , and .

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

is an important theorem in forward error correction, and describes the maximum at which reliable communication is possible over a channel that has a certain error probability or (SNR). This strict upper limit is expressed in terms of the . More specifically, the theorem says that there exist codes such that with increasing encoding length the probability of error on a can be made arbitrarily small, provided that the 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 encoding and decoding algorithms.

### Hybrid schemes

is a combination of ARQ and forward error correction. There are two basic approaches:

### Internet

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

• Each carries a CRC-32 checksum. Frames received with incorrect checksums are discarded by the receiver hardware.
• The header contains a checksum protecting the contents of the header. with mismatching checksums are dropped within the network or at the receiver.
• The checksum was omitted from the header in order to minimize processing costs in and because current technology is assumed to provide sufficient error detection (see also RFC 3819).
• has an optional checksum covering the payload and addressing information from the UDP and IP headers. Packets with incorrect checksums are discarded by the . The checksum is optional under IPv4, only, because the Data-Link layer checksum may already provide the desired level of error protection.
• provides a checksum for protecting the payload and addressing information from the TCP and IP headers. Packets with incorrect checksums are discarded within the network stack, and eventually get retransmitted using ARQ, either explicitly (such as through ) or implicitly due to a .

### Deep-space telecommunications

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 from 1968 digital error correction was implemented in the form of (sub-optimally decoded) convolutional codes and Reed–Muller codes. The Reed–Muller code was well suited to the noise the spacecraft was subject to (approximately matching a ), and was implemented at the Mariner spacecraft for missions between 1969 and 1977.

The and missions, which started in 1977, were designed to deliver color imaging amongst scientific information of and . This resulted in increased coding requirements, and thus the spacecraft were supported by (optimally Viterbi-decoded) convolutional codes that could be with an outer Golay (24,12,8) code.

The Voyager 2 craft additionally supported an implementation of a : the concatenated Reed–Solomon–Viterbi (RSV) code allowed for very powerful error correction, and enabled the spacecraft’s extended journey to Uranus and . Both craft use V2 RSV coding due to ECC system upgrades after 1989.

The 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 .

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 Earth the nature of the 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 gets larger.

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

Overview

• 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 , and 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 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 in 1951. The “Optimal Rectangular Code” used in group coded recording tapes not only detects but also corrects single-bit errors. Some , particularly , include a checksum (most often ) to detect corruption and truncation and can employ redundancy and/or to recover portions of corrupted data. Reed Solomon codes are used in 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. systems use a variety of error correction techniques to correct errors when a hard drive completely fails. Filesystems such as or , as well as some implementations, support data scrubbing and resilvering, which allows bad blocks to be detected and (hopefully) recovered before they are used. The recovered data may be re-written to exactly the same physical location, to spare blocks elsewhere on the same piece of hardware, or to replacement hardware.

### Error-correcting memory

memory may provide increased protection against by relying on error correcting codes. Such , known as ECC or EDAC-protected memory, is particularly desirable for high fault-tolerant applications, such as servers, as well as deep-space applications due to increased .

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 (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.

In addition to hardware providing features required for ECC memory to operate, usually contain related reporting facilities that are used to provide notifications when soft errors are transparently recovered. An increasing rate of soft errors might indicate that a module needs replacing, and such feedback information would not be easily available without the related reporting capabilities. An example is the ‘s EDAC subsystem (previously known as bluesmoke), which collects the data from error-checking-enabled components inside a computer system; beside collecting and reporting back the events related to ECC memory, it also supports other checksumming errors, including those detected on the .

A few systems also support .