Error detection and correction
In information theory and coding theory with applications in computer science and telecommunications, error detection and correction 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
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
In classical antiquity, copyists of the Hebrew Bible were paid for their work according to the number of stichs. 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. This also helped ensure accuracy in the transmission of the text with the production of subsequent copies. 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. Standards became such that a deviation in even a single letter in a Torah scroll was considered unacceptable. 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.The modern development of error correction codes is credited to Richard Hamming in 1947. A description of Hamming's code appeared in Claude Shannon's A Mathematical Theory of Communication and was quickly generalized by Marcel J. E. Golay.
Principles
All error-detection and correction schemes add some redundancy 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, which are derived from the data bits by some encoding algorithm. If 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. If error correction is required, a receiver can apply the decoding algorithm to the received data bits and the received check bits to recover the original error-free data. 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, and is most notably used in the Internet. An alternate approach for error control is hybrid automatic repeat request, which is a combination of ARQ and error-correction coding.
Types of error correction
There are three major types of error correction:Automatic repeat request
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, 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.
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
is a process of adding redundant data such as an error-correcting code to a message so that it can be recovered by a receiver even when a number of errors 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, as well as for reliable storage in media such as flash memory, hard disk and RAM.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 are relatively new constructions that can provide almost optimal efficiency.
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 that are both optimal and have efficient encoding and decoding algorithms.
Hybrid schemes
is a combination of ARQ and forward error correction. There are two basic approaches:- Messages are always transmitted with FEC parity data. 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.
- Messages are transmitted without parity data. If a receiver detects an error, it requests FEC information from the transmitter using ARQ and uses it to reconstruct the original message.
Types of error detection
Error detection is most commonly realized using a suitable hash function. 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.
Minimum distance coding
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
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, the four-bit block can be repeated three times, thus producing. If this twelve-bit pattern was received as – 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. The advantage of repetition codes is that they are extremely simple, and are in fact used in some transmissions of numbers stations.
Parity bit
A parity bit is a bit that is added to a group of source bits to ensure that the number of set bits 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 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.