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`Information theory` defines `coding schemes`. There are different types of coding schemes, but let us focus on coding schemes that operate on binary strings. A coding scheme is a function that maps information encoded as a string of `m` bits into a string of `n` bits. The simplest coding scheme is the (even) parity coding. This coding scheme takes an `m` bits source string and produces an `m+1` bits coded string where the first `m` bits of the coded string are the bits of the source string and the last bit of the coded string is chosen such that the coded string will always contain an even number of bits set to `1`. For example :
 
`1001` is encoded as `10010`
 
`1101` is encoded as `11011`
 
This parity scheme has been used in some RAMs as well as to encode characters sent over a serial line. It is easy to show that this coding scheme allows the receiver to detect a single transmission error, but it cannot correct it. However, if two or more bits are in error, the receiver may not always be able to detect the error.
 
Some coding schemes allow the receiver to correct some transmission errors. For example, consider the coding scheme that encodes each source bit as follows :
 
`1` is encoded as `111`
 
`0` is encoded as `000`
 
For example, consider a sender that sends `111`. If there is one bit in error, the receiver could receive `011` or `101` or `110`. In these three cases, the receiver will decode the received bit pattern as a `1` since it contains a majority of bits set to `1`. If there are two bits in error, the receiver will not be able anymore to recover from the transmission error.
 
To understand `error detection codes`, let us consider two devices that exchange bit strings containing `N` bits. To allow the receiver to detect a transmission error, the sender converts each string of `N` bits into a string of `N+r` bits. Usually, the `r` redundant bits are added at the beginning or the end of the transmitted bit string, but some techniques interleave redundant bits with the original bits. An `error detection code` can be defined as a function that computes the `r` redundant bits corresponding to each string of `N` bits. The simplest error detection code is the parity bit. There are two types of parity schemes : even and odd parity. With the `even` (resp. `odd`) parity scheme, the redundant bit is chosen so that an even (resp. odd) number of bits are set to `1` in the transmitted bit string of `N+r` bits. The receiver can easily recompute the parity of each received bit string and discard the strings with an invalid parity. The parity scheme is often used when 7-bit characters are exchanged. In this case, the eighth bit is often a parity bit. The table below shows the parity bits that are computed for bit strings containing three bits.
 
3 bits string
 
Odd parity
 
Even parity
 
000
 
1
 
0
 
001
 
010
 
100
 
111
 
110
 
101
 
011
 
The parity bit allows a receiver to detect transmission errors that have affected a single bit among the transmitted `N+r` bits. If there are two or more bits in error, the receiver may not necessarily be able to detect the transmission error. More powerful error detection schemes have been defined. The Cyclical Redundancy Checks (CRC) are widely used in datalink layer protocols. An N-bits CRC can detect all transmission errors affecting a burst of less than N bits in the transmitted frame and all transmission errors that affect an odd number of bits. Additional details about CRCs may be found in [Williams1993]_.
 
It is also possible to design a code that allows the receiver to correct transmission errors. The simplest `error correction code` is the triple modular redundancy (TMR). To transmit a bit set to `1` (resp. `0`), the sender transmits `111` (resp. `000`). When there are no transmission errors, the receiver can decode `111` as `1`. If transmission errors have affected a single bit, the receiver performs majority voting as shown in the table below. This scheme allows the receiver to correct all transmission errors that affect a single bit.
 
Received bits
 
Decoded bit
 
Other more powerful error correction codes have been proposed and are used in some applications. The `Hamming Code <https://en.wikipedia.org/wiki/Hamming_code>`_ is a clever combination of parity bits that provides error detection and correction capabilities.
 
Reliable protocols use error detection schemes, but none of the widely used reliable protocols rely on error correction schemes. To detect errors, a frame is usually divided into two parts :
 
a `header` that contains the fields used by the reliable protocol to ensure reliable delivery. The header contains a checksum or Cyclical Redundancy Check (CRC) [Williams1993]_ that is used to detect transmission errors
 

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../../principles/reliability.rst:431 • ../../principles/reliability.rst:432 • ../../principles/reliability.rst:433 • ../../principles/reliability.rst:434 • ../../principles/reliability.rst:435 • ../../principles/reliability.rst:436 • ../../principles/reliability.rst:437 • ../../principles/reliability.rst:438 • ../../principles/reliability.rst:452 • ../../principles/reliability.rst:453 • ../../principles/reliability.rst:454 • ../../principles/reliability.rst:455
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