Hamming's codes which by examples of the few known perfect codes belong to cyclic codes. They have code distance of d=3 and correct all single errors. Among cyclic codes broad application was found by Boaz's codes - Chowdhury - Hokvingem (BChH).
Thus, the generating matrix (contains all information, for coding. It has to in memory of the coding device. For a binary code memory size is equal to kXn of binary symbols. At a tabular task of a code the coding device has to remember
Among razdelimy codes distinguish linear and nonlinear. Codes in which the digit-by-digit sum on the module of 2 any two code words is also the code word belong to the linear. The linear code is called systematic if the first k of symbols of its any code combination are, other (n-k) of symbols — test.
Follows from definition that any linear code (, k) to receive from k of linearly independent code combinations their posimvolny summation on the module 2 in various combinations. Initial linearly independent code are called basic.
Probabilistic methods of decoding are rather difficult in realization, though provide a high. Along with them widely apply simpler. For this purpose use class SK, allowing threshold decoding.
Linear (, k) the code can be set by a matrix of N of dimension (rkhp). Thus the combination In belongs to a code only if a vector In to all lines of a matrix of N i.e. if equality is carried out (
It is known that channels on which information is transferred, practically never happen ideal (channels without hindrances). At them almost always there are hindrances. Difference only in the level of hindrances and their spectral structure. Hindrances in channels are formed for various reasons, but the result of their impact on the transmitted data always one – information is lost distorted).
It is obvious that as a polynomial of b (x) it is possible to work a (x) (x). However thus information and test symbols are mixed that decoding process. Therefore the following method of finding of a polynomial of b is generally put into practice (x).
where I-single kXk a submatrix, P-kX(n-k) - the submatrix of test symbols defining properties of a code. The matrix sets a systematic code. It is possible to show that for a linear code there is an equivalent systematic code.
Process of coding can be presented as multiplication of a polynomial of an entrance information of u (D) by the generating G(j)(D) code polynomials, describe communications of cells of the register of the coder with its exits (:
Let's notice that lines of a test matrix the linearly. Therefore the test matrix can be used in a generating for some other linear code (, the isthmus) called by the dual.
The polynomial has the following structure: the first n-k of members of the lowest order are equal to zero, and coefficients of the coincide with the corresponding coefficients of an polynomial and (x). The polynomial with (x) has degree to n-k. Thus, in the found polynomial b (x) coefficients at x in n-k degree coincide with symbols, and coefficients at other members, a polynomial with above (x), coincide with test. On the basis of the provided schemes of multiplication and division of the coding devices for cyclic codes are also under construction.
Example of a nonlinear code is Berger's code, at test symbols present binary record of number of units to sequences of information symbols., such is a code: 00000; 00101; 01001; O111O; 10001; 10110; 11010; 1111 Codes of Berger are applied in channels. In symmetric channels they find all single mistakes and some part of the repeated.
which are called as the check equations. From (1 follows that test symbols of code combinations of a linear code are formed by various linear combinations of symbols. Units in any j-y to a line of the submatrix P entering a test matrix (1, specify, what information symbols participate in formation of j-go of a symbol.
It is obvious that linear (, k) a code it is possible to construct, the check equations (Thus the first k of symbols of a combination information, and the others the isthmus of - test, formed according to (
Continuous codes are characterized by that operations of a and decoding are made over a continuous of symbols without splitting it on blocks. Among the svertochny codes are most applicable.