Coding theory

The coding theory, the mathematical theory of error-detecting and correcting codes. Such codes are used in all cases where application of digital data to be protected from occurring in transmission or storage errors. Examples are to communicate with objects in space and the storage of data on a CD.

Large parts of the coding theory based on the algebra, also often why the term is used algebraic coding theory to draw a clear boundary to the related information theory. In addition to the algebra used in coding theory and methods from combinatorics, number theory and finite geometry used.

History

As the founder of algebraic coding theory are Marcel JE Golay, in 1949 the comprehensive half page article Notes on digital coding published, as well as Richard Hamming with its patent law in 1950 delayed work published error -detecting and error -correcting codes.

Description

From cryptography and data compression, coding theory differs in its objective: The former have data against unintended receiver secured or the amount of data to be reduced, it is of interest in coding theory, to increase the amount of data deliberately by inserting redundancies to thereby achieving a hedge against errors. These types of data modification can also be combined with each other: It is not uncommon that data is compressed first, then cryptographically encrypted and will eventually be coded against transmission errors. Specifically, the upstream connection of a compression method is often applied, since this data in a statistical distribution of characters is produced, benefiting from the encoding.

Important parameters of a code, the information rate ( a parameter for the information contained in a fixed amount of data amount of information ), and the correction rate ( a parameter for the error-resistance at a fixed amount of data ). In addition to codes with good information and correction rate it is usually also interested that require the encoding and decoding algorithms do not have high technical requirements. It is currently not possible to optimize all these properties simultaneously. It must therefore be always chosen new to the practice, which code provides the best compromise for a particular application.

For simple algorithms for encoding and decoding, it is helpful to the Code impose a rich algebraic structure as possible. The theoretical treatment of such codes is easier than in the general case. Against this background, the group codes are ( structure of a group ), the linear code ( structure of a finite dimensional vector space ), and originated the cyclic code ( structure of a finite algebra). Also the investigation of the associated group ring over a finite field are often further insight into the structure of the code. Another class of codes are convolutional codes.