Theory of the codes

In Information theory, the theory of the codes treats codes and thus of their properties and their aptitudes to be been useful on different channels of communication. One distinguishes two models from communication: with and without Noise. Without noise, the Codage of source is enough with the communication. With noise, the communication is possible with the correct codes.

History

By defining the Information in a mathematical way , the stage-key with the foundation of the theory of the codes was crossed by Claude Shannon. Other definitions exist, but the Entropie of Shannon was most prolific. Thus, one is ready to answer the two fundamental questions of the Information theory: which are the resources necessary to the transmission of information and the quantity of information which one can transmit in a reliable way.

It is of this last question of the Codage of channel covered by the theory of the codes. While answering the two basic questions of the information theory, Shannon precisely did not provide a very powerful whole of correct codes. In particular, it did not determine an example of code which reaches the limit envisaged by its theorem of the coding of channel.

It is this vacuum which the theory of the codes fills. There exists a multitude of methods nowadays aiming at producing good correct codes.

Properties of the codes

One distinguishes initially the codes by the quantity from information transmitted by a Symbole. Considering the symmetrical binary Canal is most common, one will often consider a binary Code. There exist however also tertiary codes and, in general, codes Q-surfaces.

The following names of variables, are used most of the time by convention. $C$ is a code containing $M$ words of code, i.e., of dimension Mr. the length of a word of code is indicated by $n$ . Such a code is known as code $(N, M)$ .

Detection and correction of errors

The majority of the codes are used is for the detection or the correction of error.

Minimal distance and decoding

The minimal distance from a code influences the probability of error of decoding. The minimal distance is an important parameter, indicated $d$ . Such a code is known as code $(N, M, d)$ .

Families of codes

Equivalent codes

Two codes are equivalent if all their properties of correction of error are the same ones.

Types of codes

One generally distinguishes three types of codes.

a linear Code is a correct Code on which one imposes a structure of vector Space.

a cyclic Code is a correct code which has more structure that of a vector space. In general, one can conceive the words of code like Polynôme S. Exemples: the Reed-Solomon code and the Hamming code.

a nonlinear Code is a correct code which can be built of a variety in ways, without obtaining the structure of a vector space.

There is a small number of special cases. A commonplace code is a code which recopies the initial message literally, from where its Trivialité. A systematic code is a code which enumerates all the possible messages and their corresponding encoding.

In addition, certain correct codes can be used like quantum codes.

Other types of important codes are:

algebraic Code

random Code

Families

The correct codes can also be classified by families.

a code of parity adds one or more bits of parity to the message.

a Code of repetition sends several copies of each bit to being transmitted.

the Hamming codes train the most known family. The binary Hamming codes are equivalent to the cyclic codes and certain nonbinary are it too.

a Code of Golay is a linear Code considered important in theory and in practice.

a Code of Reed-Müller is a linear code whose properties of decoding are considered particularly practical.

the codes BCH are a generalization of the Hamming codes. They also acts of cyclic codes. A particular case is the Reed-Solomon code.

a quadratic Code of residue is a cyclic code based on the quadratic Résiduosité.

a Code of Goppa which, like the cyclic codes, is based on a polynomial, said polynomial of Goppa.

a stabilizing Code is based to the measure of a syndrome, i.e. of a Vecteur in $F_2^ {n-k}$ .

a Code expandor is a linear code with which it is always possible to correct a constant fraction of errors.

the codes superconcentrator of Spielman are the only codes being able to be coded and decoded in linear time.

a Code alternating is a linear code of practical importance.

a Code of Hadamard is a code generated starting from a Matrice of Hadamard.

Coded representations

One can obtain new codes starting from operations which combine one or two basic codes.

commonplace operations: trouage and copies
concatenation: Code of Forney, Code of Justesen
produced

Other properties

One distinguishes also certain classes from codes by their properties.

Code intersecting
Code separating
Code MDS

Code and “design”

There is a connection between the codes and the combinatoriaux designs.

The principal problem of the theory of the codes

That is to say $A_q (the N, d)$ largest $M$ for which there exists a code $(N, M, d)$ and $q$ -naire. The principal problem of the theory of the codes is to determine these values.

Coding of source

The goal of the coding of source can be to compress the repetitive information of the language, its Redondance. For any language, one can consider the Entropie of a message, i.e. the quantity of transmitted information. This gives place to the Théorème of the coding of source.

Coding of channel

The goal is to add repetitive information to a message to compensate for the noise on the channel of communication. This gives place to the Théorème of the coding of channel and it is with this one that one owes the origin of the theory of the codes .

Certain cryptographic problems are based on the assumption of the difficulty of decoding.

Algebraic theory of the codes

The algebraic theory of the codes is a under-field of the theory of the codes where the properties of the codes are expressed algebraically. In other words, the approach is algebraic in opposition to the traditional approach which is probabilist. One studies there mainly:

the construction of “good” codes, i.e. with certain desirable parameters, such:

the length of the words of code
the full number of valid words of code
the Distance Hamming minimal between two valid words of code

the effective decoding of these codes

References

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