Multidimensional parity-check code

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A multidimensional parity-check code (MDPC) is a simple type of error correcting code that operates by arranging the message into a multidimensional grid, and calculating a parity digit for each row and column. In general, an n-dimensional parity scheme can correct n/2 errors.[ citation needed ]

A parity bit, or check bit, is a bit added to a string of binary code to ensure that the total number of 1-bits in the string is even or odd. Parity bits are used as the simplest form of error detecting code.

Contents

Example

The two-dimensional parity-check code, usually called the optimal rectangular code, is the most popular form of multidimensional parity-check code.

Assume that the goal is to transmit the four-digit message "1234", using a two-dimensional parity scheme. First the digits of the message are arranged in a rectangular pattern:

12
34

Parity digits are then calculated by summing each column and row separately:

123
347
46

The eight-digit sequence "12334746" is the message that is actually transmitted. If any single error occurs during transmission then this error can not only be detected but can also be corrected as well. Let us suppose that the received message contained an error in the first digit. The receiver rearranges the message into the grid:

923
347
46

The receiver can see that the first row and also the first column add up incorrectly. Using this knowledge and the assumption that only one error occurred, the receiver can correct the error. In order to handle two errors, a 4-dimensional scheme would be required, at the cost of more parity digits.

Decoder

An n-dimensional parity scheme is only guaranteed to correct up to n/2 errors, as the minimum distance is (n + 1). As with all block codes, a soft-decision decoder may be able to correct more than this.

In coding theory, block codes are a large and important family of error-correcting codes that encode data in blocks. There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract definition of block codes is conceptually useful because it allows coding theorists, mathematicians, and computer scientists to study the limitations of all block codes in a unified way. Such limitations often take the form of bounds that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors.

In information theory, a soft-decision decoder is a kind of decoding methods – a class of algorithm used to decode data that has been encoded with an error correcting code. Whereas a hard-decision decoder operates on data that take on a fixed set of possible values, the inputs to a soft-decision decoder may take on a whole range of values in-between. This extra information indicates the reliability of each input data point, and is used to form better estimates of the original data. Therefore, a soft-decision decoder will typically perform better in the presence of corrupted data than its hard-decision counterpart.

See also

In information theory and coding theory with applications in computer science and telecommunication, 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.

In telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable or noisy communication channels. The central idea is the sender encodes the message in a redundant way by using an error-correcting code (ECC).

In information theory, a low-density parity-check (LDPC) code is a linear error correcting code, a method of transmitting a message over a noisy transmission channel. An LDPC is constructed using a sparse bipartite graph. LDPC codes are capacity-approaching codes, which means that practical constructions exist that allow the noise threshold to be set very close to the theoretical maximum for a symmetric memoryless channel. The noise threshold defines an upper bound for the channel noise, up to which the probability of lost information can be made as small as desired. Using iterative belief propagation techniques, LDPC codes can be decoded in time linear to their block length.

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