RU2693190C1 - Method of diagnosing non-binary block codes - Google Patents

Abstract

FIELD: electrical communication engineering.

SUBSTANCE: invention relates to communication engineering and can be used to determine an unknown coding structure of non-binary systematic codes and non-systematic codes based on analysis of the received code sequence. Method comprises steps of: storing N code blocks, extracting pairs of code blocks which are pairwise compared and folded, at that, maximum degrees of members of selected polynomials describing compared code blocks are compared, a lower polynomial is shifted bit by bit, all coefficients of each polynomial are divided by a coefficient of its maximum degree, comparing a larger polynomial with a shifted smaller polynomial, if they are equal, the form of the smaller polynomial before shift is stored, if they are unequal are added, coefficients in the Galois field are added for members with the same degrees of polynomials and returned to comparison of smaller polynomial and obtained result of addition in Galois field, form of obtained polynomials is memorized, maximum number of obtained identical results of comparison and addition is selected.

EFFECT: high noise-immunity of transmitting information.

1 cl, 3 dwg

Description

A method for diagnosing non-binary block codes relates to communication technology and can be used to determine the unknown coder structure of non-binary block systematic codes, including Reed-Solomon codes, and non-systematic codes based on an analysis of the received code sequence.

There are various methods of block coding and decoding used to correct errors that occur during transmission, and described, for example, in books: V. Ipatov, V. Orlov, I. Samoilov. Mobile communication systems - M .: Hotline-Telecom, 2003; Morelos-Zaragoza R. The art of robust coding. Methods, algorithms, application - M: Technosphere, 2006; Sklyar B. Digital communication. Theoretical foundations and practical application - M: Ed. House "William", 2003.

They describe the operations of obtaining a code block from the input information sequence of characters by performing convolution operations of a certain number of information symbols with polynomial generators (generating polynomials) and the sequential transmission of similar code blocks. At the same time, to implement a non-systematic code, the group of consecutive source information symbols is multiplied by the generator polynomial, represented as a vector with binary elements. To implement a systematic code, the original group of information symbols is shifted by a certain number of binary digits. Further, the result of the shift is divided into the generating polynomial and the quotient is added modulo 2 with the shifted group of symbols. After reception, the sequence is decoded by one of the possible methods, when decoding, the entire received code block is divided into the generating polynomial, and the result of the division is used to localize and correct errors.

The disadvantage of the described method is the necessity of knowing on the receiving side the correct type of the generating polynomial used. If it is unknown, for example, lost or damaged, then error correction decoding is impossible and the correcting ability of the code cannot be implemented. This leads to a decrease in noise immunity of information transfer.

The closest to the claimed is the method according to the patent of the Russian Federation No. 2631142 on "Method for diagnosing cyclic codes" on application No. 2016107245, priority from 02/29/2016, registered in the state register of inventions September 19, 2017, IPC H04L 17/00, authors Korneev N.N. , Polushin PA, Nikitin O.R.

The prototype method is designed to diagnose binary block codes. In the prototype method, the received N code blocks are memorized, further extraction of pairs of code blocks, pairwise comparison and addition of blocks, memorization of the type of polynomials and selection of the maximum number of results, the pairwise comparison and addition of blocks operation consists of sequential polynomial comparison operations, bitwise shift of the polynomial , comparing the result with the shifted polynomial and bitwise addition modulo 2.

Using the operations of the method, a common polynomial is determined for each pair of code blocks analyzed and its type is remembered. After considering all the code blocks, the polynomial, the maximum number of times stored in memory, is selected as the diagnostic result.

The main disadvantage of the prototype is that in non-binary codes each character is represented not by a single bit, but by several bits (m bits), and is described not by a binary number, but by special coefficients — objects of a finite Galois field algebra (see, for example, the books mentioned B. Sklyara or R. Morelos-Zaragoza). The rules of ordinary binary Boolean algebra are not applicable to these coefficients, and the operations of addition, multiplication, etc., although they are based on the logical processing of groups of binary bits that make up each character, but must be performed on the basis of special rules generated using the so-called. primitive polynomials. Thus, all prototype operations performed on the basis of Boolean algebra are unsuitable for processing non-binary characters.

In addition, when the higher degrees of binary polynomials are aligned for the operation of pairwise addition modulo 2, then in the method of the prototype the highest terms of the polynomials are always subtracted and disappear, and the maximum degree of polynomials decreases. In the case of nonbinary polynomials, the coefficients of the higher polynomial can have not two valid values (0 and 1), but much more than possible values. Therefore, in the general case, direct subtraction of polynomials cannot reduce the maximum degree of a polynomial, and the principle implemented in the prototype method cannot be realized for non-binary codes.

The required level of noise immunity of information transmission using the correcting capabilities of decoders in the absence of information about the encoder is not ensured, since the performance of decoders is impaired.

The objective of the proposed method for the diagnosis of non-binary block codes is to determine the structure of the used non-binary coder based on the analysis of the received code blocks to ensure the performance of decoders and improve the noise immunity of information transmission.

The problem is solved in that a method for diagnosing non-binary block codes, including storing N code blocks, extracting pairs of blocks, pairwise comparison and addition of blocks, a bitwise polynomial shift, remembering the type of polynomial and choosing the maximum number of results, introduces a comparison of the degrees of polynomials, division the coefficients at the highest degree of a polynomial, the definition of equality of polynomials and the addition of polynomials in the Galois field, while first remember N code blocks, then remove pairs of code blocks, then code blocks are compared and put in pairs, then the type of polynomials obtained as a result of comparison and addition is memorized, after which the maximum number of identical results of comparison and addition obtained is chosen, and when pairwise comparison and addition of code blocks, the maximum powers of the members of the selected polynomials describing the compared code blocks are compared, then shift the smaller polynomial bitwise, then divide all the coefficients of each polynomial by the coefficient of its member of the maximum degree, then I compare greater shifted polynomial by polynomial smaller, in case of equality of the polynomial to form a smaller shear stored, in case of inequality is folded in the Galois field coefficients of the terms of the same degrees of polynomials and to return smaller than the polynomial and the result of addition in the Galois field.

The drawings show: in FIG. 1 is a schematic sequence of operations of the proposed method; in fig. 2 is an example of a block diagram of a device for implementing the proposed method; in fig. 3 shows an example of the implementation of registers for storing codeword symbols.

FIG. 1 are designated: 1 - storing of N code blocks; 2 - extraction of pairs of blocks; 3 - remembering the type of polynomials; 4 - selection of the maximum number of results; 5 - comparison of the degrees of polynomials; 6 - bitwise polynomial shift; 7 - division by coefficients at the highest degree of the polynomial; 8 - definition of equality of polynomials; 9 - addition of polynomials in the Galois field; 10 - pairwise addition and comparison of blocks.

FIG. 2 presents: a block of addition in the Galois field 11; the first 12, the second 13, the third 14 and the fourth 15 switches; the first 16 and second 17 comparison blocks; the first 18 and second 19 determinants of the maximum degree of a polynomial; the first 20, second 21, third 22 and fourth 23 registers; subtractor 24; control unit 25; block allocation of the maximum amount of 26; shift register 27, the first 28 and the second 29 dividers; the first 30 and second 31 memory blocks.

FIG. 3 presents: a register 32 for storing codeword symbols; binary cells 33 for storing the bits constituting the codeword symbols.

The operation of the proposed method are as follows. During operation 1, N received code blocks are memorized. At the same time, it is possible to further retrieve from memory any stored code blocks. Next, two different code blocks from the stored ones are compared in pairs. With the number of N blocks, N (N-1) / 2 different combinations of pairs of different blocks are possible. These blocks are retrieved from memory by pairwise operation 2 (the sequence of extracting the value does not play) and then processed by the general operation 10 of pairwise comparison and addition of blocks.

, где через X обозначается сдвиг по времени на длительность одного символа, через X² сдвиг по времени на длительность двух символов, через X³ - трех символов, и т.д., при этом максимальная степень полинома на единицу меньше количества двоичных разрядов кодового блока. Коэффициенты a _i при членах полинома различных степеней являются элементами поля Галуа, и действия над ними подчиняются правилам конечной алгебры (см., например, упомянутые книги Б. Скляра или Р. Морелоса-Сарагосы).Operation 10 pairwise addition and comparison of blocks consists of several operations. As you know, each code block is a sequence of n characters, each character consists of m binary bits, where m is an integer selected during encoding. A code block can be described as a polynomial,

where X denotes the time shift for the duration of one character, X ² for time duration for two characters, X ^{3 for} three characters, etc., with the maximum degree of the polynomial one less than the number of binary digits of the code block . The coefficients a _i with terms of polynomials of different degrees are elements of the Galois field, and the actions over them obey the rules of finite algebra (see, for example, the books mentioned by B. Sklyar or R. Morelos-Zaragoza).

In operation 5 of the comparison of polynomials, the maximum degrees are compared with the higher terms of the polynomials describing the two selected code blocks. At the same time, it is determined which of them is larger than the other (the maximum degree of which is greater than the block) and by how many digits. The compared two code words y ₁ and y ₂ can be described as polynomials:

where n and m are the numbers of maximum digits in the first and second compared code words containing one in the binary record of the code word, counting from the first digit; X denotes a delay per clock (for the duration of one character); the coefficients a _i and b _i are elements of the Galois field and can take values in the range from zero to 2 ^m -1.

.Operation 5 determines the degrees n and m. Let the first polynomial y ₁ be greater. In operation 6 of the bitwise shift of the smaller polynomial, the polynomial y _{2 is} shifted upward by a certain number of digits. The number of digits for which the shift is made is equal to the difference of their maximum degrees, i.e. equals nm. As a result of the shift, the polynomial X ^nm y ₂ is obtained. Thus, if m> m, then such a shift is made, if n = m, (although the polynomials are different), they do not shift. After such a shift, the second polynomial takes the form

.

In operation 7, all coefficients with terms of both polynomials of smaller degrees are divided into coefficients with their highest powers, i.e. for the polynomial y ₁ at a _n , for the polynomial y ₂ for b _m . After that, the higher terms of both polynomials become the same and equal to X ⁿ .

In the next operation 8, it is determined whether the polynomials have become completely equal after the previous operation 7. If these polynomials become exactly equal to each other (all coefficients with members of all degrees are equal), i.e. are the same, then the processing cycle of this pair of code blocks is terminated, and at operation 3 a smaller polynomial is supplied before its shift. If they remain not equal to each other, then operation 9 produces the bitwise addition of polynomials in the Galois field according to the rules of finite algebra (the addition of coefficients with terms with the same powers). According to the rules of finite algebra, the addition of polynomials is equivalent to the subtraction of coefficients for members of the same degree. Thus, after operation 9, the higher terms of both polynomials disappear, and the result of addition always has a lesser degree than the added polynomials.

After this, the processing returns to operation 5, where other polynomials are already compared. One of them is a polynomial - the result of the addition made in the Galois field by the operation 9 (subtraction of the higher terms of the polynomials). Another comparable polynomial is the smaller polynomial before its shift. Then follow the described operations again. Thus, for the selected pair of source code blocks, in the general case, the operations 5-6-7-8-9 are repeated several times. The type of polynomials being analyzed changes with each repetition. With each repetition, the polynomials themselves necessarily decrease, and the degree of the maximum polynomial decreases. This happens until operation 8 fixes the equality of the polynomials being analyzed. After that, the smaller polynomial obtained before the shift is remembered by operation 3, and operation 10 as a whole proceeds to the analysis of the next pair of code blocks.

After each completion of operation 10 as a whole (pairwise comparison and addition of blocks) in operation 3 (memorizing the type of polynomials), the result of operation 10 is memorized. If the previous result of operation 10 previously remembered the same result (the same polynomial) related to another a pair of compared code blocks, this fact is remembered and increases the amount accumulated by the previous results related to the polynomial of this obtained type. The number of received results of each type is calculated.

Operations 10 pairwise comparison and addition of blocks is performed on all the different N blocks, which are stored in operation 1.

The number of such pairwise treatments is N (N-1) / 2. When all these pairings are completed, as a result of performing operation 3, all detected types of common polynomials in all combinations of polynomials in pairs turn out to be remembered. After completing all N (N-1) / 2 comparisons, operation 4 of selecting the maximum number of results determines the most frequently memorized results, i.e. what kind of polynomials are more memorized. (If, for example, every fact of memorization is done in the form of adding one to the already available amount in a memory cell related to a polynomial of this type, then in fact, this operation determines which polynomial memorized by procedure 3 corresponds to the largest memorized amount.) This polynomial is a set of coefficients for polynomial terms of different degrees, which determines the generator polynomial that is used in the transmitter to encode the analyzed signal. It is displayed further as the final result of the diagnostic procedure.

The blocks of the device in FIG. 2., as an example of the implementation of the proposed method, work as follows. The first memory block 30 receives received code blocks in the form of sequences of characters. It stores N different code blocks. (The sequence of the memorized blocks does not matter, it can be either successively following blocks, or arranged in other ways.)

Different blocks are extracted from this memory block, one of these blocks is fed to the input of the first switch 12, another block is fed to the input of the second switch 13. The sequence of extracting pairs of blocks from the memory has no value, the current numbers of the blocks to be extracted are set by the control unit 25. In the block Memory 30 is loaded with N code blocks, so by performing N (N-1) / 2 cycles, it is possible to analyze N (N-1) / 2 different pairs of blocks. Each cycle produces the same set of actions.

The analysis cycle of the next pair of code blocks is as follows. At the very beginning of the cycle, the first 12 and second 13 switches connect the first and second multichannel outputs of the first memory block 30 to the multichannel parallel inputs, respectively, of the first 20 and second 21 registers, where the pair of code blocks currently being analyzed is written in the form of a sequence of characters, each of which consists of t bits.

In the first comparison block 16, the degrees are compared with the highest term of the polynomial of each code block (senior degrees) recorded in the first 20 and second 21 registers. The output of this first comparison unit controls the third switch 14.

If the higher degrees of the polynomials of code blocks recorded in registers 20 and 21 are not equal to each other, then the first comparison block 16 uses the third switch 14 to supply the code block with a higher higher power to the input of the first determinant of the maximum degree 18 and to the parallel input of the third register 22 , and the code block with a lower high power is supplied by the switch 14 to the input of the second determinant of maximum degree 19 and to the parallel input of the fourth register 23. The signals received by the registers are written to these registers. The polynomial recorded in the fourth register 23 is also overwritten in the shift register 27.

If it turns out that the highest powers written in registers 20 and 21 polynomials are equal to each other, then using the third switch 14, the input of the first determinant of the maximum degree of the polynomial 18 sends a signal from the output of the first register 20, and the input of the second determinant of the maximum degree of the polynomial 19 the signal from the output of the second register 21.

The determinants of maximum degree 18 and 19 define the orders of polynomials, i.e. maximum degrees of X.

Next, in the subtractor 24, the arithmetic difference of their orders (the difference of the maximum degrees of the polynomials) is found, and based on the output of the subtractor in the shift register 27, the entire polynomial recorded in the fourth register 23 is shifted towards an increase in the degree by the resulting value of this arithmetic difference. Thus, after this shift, the orders (maximum degrees) of the polynomials recorded in the third register 22 and in the shift register 27 become the same.

Further, in the first 28 divisor, all coefficients are divided at the members of the first polynomial by the coefficient for the highest power of this polynomial, and in the second 29 divider, all coefficients are divided for the terms of the second polynomial by the coefficient for the highest power of this polynomial. These polynomials are fed to the second comparison block 17 and to block 11, where coefficients are added for the same degrees of polynomials in the Galois field according to the rules of finite algebra.

In the second comparison block 17, the polynomials recorded in registers 22 and 28 are compared. If they are completely equal to each other (ie, they are the same), the second comparison block reports this to the control unit 25. Then this cycle ends, the control block opens the fourth the switch 15 and the signal from the output of the fourth register 23 is supplied to the second memory block 31. In this second memory block, each possible type of polynomial corresponds to its own memory cell. Initially, before the start of the diagnostic procedure, all cells contain zeros. When the next polynom is defined in the next cycle, then in the cell corresponding to it, one is added to the amount that was already written there.

If the second comparison block 17 did not fix the equality of polynomials in the third and fourth registers, the result of the addition in block 11 is fed to another input of the first switch 12, and to another input of the second switch 13 a polynomial written in the fourth register 23 is fed. After that, the outputs of both switches Signals from the first memory block 30 are not connected, but newly received signals from their other inputs. After that, the whole sequence of actions is repeated. The operation of switches, registers and memory blocks is controlled by the control unit 25.

When all N (N-1) / 2 combinations of the considered code blocks have been enumerated, in the second memory block 31, a different number of units are recorded in different cells. After that, the block for allocating the maximum amount 26 determines in which cell the maximum number is written. The polynomial corresponding to this cell is fed to the output as the final result of the entire diagnostic procedure.

Register 32 in FIG. 3 consists of n groups in which there are successively located m cells 33 for storing m bits constituting the symbol of the code word. Registers 20-23 allow parallel input and output of information on groups. The shift register 27 shifts the contents of the cells in groups of m bits.

Consider the operation of the proposed method in more detail. As you know, with the non-systematic coding, each i-th code block can be described as a product of two polynomials (y _i (X) = m _i (X) g (X), where g (X) - generator polynomial (polynomial generator) of the used encoder, common to all code words; m _i (X) is the part of the i-th code word containing the information transmitted in it. That is, all code blocks have at least one common polynomial g (X). The maximum degree of the polynomial g (X) is b. The degree of the polynomial m _j (X) depends on the information transmitted in this block and can reach a maximum value equal to k.

With systematic coding, the code block in the encoder is formed differently. For this, the initial binary information polynomial m _i (X) is initially multiplied by X ^b , which corresponds to a shift to b bits upwards. The result is a polynomial m _i (X) X ^b with a maximum degree, in general, equal to n = k + b. Further, it is divided into the polynomial generator g (X). The maximum degree of the resulting residue r _i (X) is equal to b, i.e. equal to the number of zeros in the record m _i (X) X ^b , starting with the first digit. After that, the remainder r _i (X) is added to the polynomial, forming the code block transmitted by the transmission channel y _i (X) = m _i (X) X ^b + r _i (X). Thus, in this case, too, the code block is a multiple of the polynomial g (X), i.e. the polynomial generator is one of the factors of the polynomial y _i (X), and you can write y _i (X) = _Mi (X) g (X). It is this fact that is used in the receiver to calculate error syndromes and correct them.

If we compare pairs of different code blocks in pairs, although the sets of factors into which their polynomials can be expanded will differ, the factor g (X) will be present in all code blocks. Naturally, the same factors may sometimes appear in the information part M _i (X) of different code words.

Denote the same factors in the information part of the two compared code blocks by M _C (X), and the different parts by M _d1 (X) and M _d2 (X). In another pair of code words, the part M _C (X) will generally differ from the common part in the first pair, in the third pair differ from the first two, and so on. Thus, if we compare a sufficiently large number of pairs, despite the fact that in each pair, apart from the polynomial g (X), there may be common polynomials and other polynomials, but in the totality of the code blocks being analyzed, only the sought generator will be the common factor polynomial generator g (x).

Accordingly, the operations in FIG. 1 perform the following actions. In operation 1, the N code blocks received at the receiver are stored. The number N must be sufficiently large, for example, significantly larger than the total number n of characters in the code block.

Since the information parts of different blocks in the vast majority of cases are different, then the appearance of identical factors in the information part is also unlikely, it is much more likely that the common part will consist only of a polynomial generator. A large number of N is necessary so that the probability of the appearance of the required generating polynomial in the analysis is significantly greater than other analysis results. With further increase in N, this probability increases, which makes it possible to reduce the diagnostic error to the required levels.

The appearance of erroneous symbols during transmission over a communication channel only leads to the appearance of other analysis results and the memorization of other types of polynomials. This negative phenomenon can be attenuated to an acceptable level by an increase in N. Thus, operation 1 ensures that N codewords are memorized and their pairwise transmission for subsequent processing. The number of pairs of different code blocks is N (N-1) / 2.

, а второй полином равен

, где a _S÷a ₀ и b_S÷b₀ - некоторые коэффициенты при членах полиномов, являющиеся элементами поля Галуа. Операцией 5 сравниваются величины старших степеней S и Т.Operation 10 as a whole compares and adds the code blocks fed to it. It consists of several successive operations. In operation 5, the polynomials of the two analyzed code blocks are compared, and the maximum degrees of the polynomials of both blocks are determined. Let the polynomial of one of the code blocks be

and the second polynom is

, where a _S ÷ a ₀ and b _S ÷ b ₀ are some coefficients for the terms of the polynomials that are elements of the Galois field. Operation 5 compares the values of the higher degrees S and T.

Suppose it turns out that S> T, i.e. polynomial y _{1 is} longer than polynomial y ₂ . Then, by the next operation 6 of the bitwise shift of the smaller polynomial, the second polynomial is multiplied by X ^ST , that is, the upward shift by ST degrees of all its members. If the second polynom is greater than the first, then the first polynom is multiplied by the required difference value, i.e. the degrees of all its members increase by the corresponding difference number. If the maximum degrees of both polynomials are equal (i.e. S = T), then in operation 6 no degree shift is performed. Thus, after operation 6, the maximum degrees of both polynomials will coincide.

,

, где A_S-1=a _S-1/a _S, A_S-2=a _S-2/A_S, …, B_S-1=b_S-1/b_S, B_S-2=b_S-2/b_S и т.д. Коэффициенты при членах старших степеней становятся одинаковыми и равными единице.The next operation 7 is the division of all the coefficients of the first polynomial by the value of a _S and the division of all the coefficients of the second polynomial by the value of b _T. Suppose that S> T, i.e. the higher degree of the second polynomial was less. Then the polynomials after division take the form:

,

where A _S-1 = a _S-1 / a _S , A _S-2 = a _S-2 / A _S , ..., B _S-1 = b _S-1 / b _S , B _S-2 = b _{S -2} / b _S etc. The coefficients at the members of the senior degrees become the same and equal to one.

Further, in operation 8 for determining the equality of polynomials, it is checked whether y ₁ (X) and y ₂ (X) are completely equal, i.e. compares the magnitude of the larger polynomial and the result of the shift of the smaller polynomial. If, as a result of the shift, the smaller polynomial becomes completely equal to the larger polynomial, then the general operation 10 of pairwise comparison and addition of blocks is terminated. The form of the smaller polynomial is transmitted to operation 3 and is remembered.

If the compared polynomials are not equal to each other, then operation 9 is performed there. In accordance with the rules of finite algebra, the addition of coefficients in the Galois field with terms of the same powers of both polynomials is performed. The operation of addition is equivalent to the operation of subtraction, and since their terms of the maximum degrees become the same, after the implementation of this operation, the maximum degree of difference decreases by one. Naturally, the values of the coefficients change for terms of other degrees. After this, operation 5 compares the polynomials again, but with two other polynomials, one of which is equal to the mentioned lower polynomial before its bitwise shift, and the other is the result of addition in the Galois field of operation 9. After it, the described operations are performed again, until the operation 8 the equality of the polynomials being processed will not be established.

When the procedure performed by the general operation 10 of comparing and adding polynomials ends, this results in the result that the polynomials that are common to both code blocks analyzed in it remain. Indeed, let the compared code blocks be described by polynomials: y ₁ (X) = M _d1 (X) M _C (X) g (X) and y ₂ (X) = M _d2 (X) M _C (X) g (X) .

As mentioned, addition and subtraction in the Galois field are equivalent operations, since lead to the same results. Therefore, the difference polynomial y ₃ = y ₁ −y ₂ = (M _d1 + M _d2 ) M _C g = (M _d1 –M _d2 ) M _C g = M ₃ M _C g will contain the same common factors as the original polynomials before addition, and only the parts of the polynomials that differ in y ₁ and y ₂ change.

Since before the addition (subtraction) the highest degree of the smaller polynomial was temporarily equated to the highest higher, after this operation the highest degree of the result of addition is always reduced by one compared to the maximum order of the polynomials being analyzed. If after it the result is not equal to the smaller of the current polynomials, then the alignment of the higher powers of the two analyzed polynomials and their subtraction are repeated. The order of the differing part (M ₃ ) is again reduced by one, and so on. Thus, after conducting a certain number of repetitive operations, 5–9 polynomial M ₃ becomes equal to unity. The number of repetitive operations is determined by the specific type of polynomials M _d1 and M _d2 and is equal to the order of their maximum.

Let us demonstrate this procedure for determining identical divisors for a pair of code blocks using a specific example. As is known, the coefficients in the Galois field can be numbered differently. In the above books, numbering is applied in the form of the degrees of the base element a . For the convenience of specific calculations, index numbering is also used for the base element a , for example, in the Matlab computing environment (where a ₀ = 0 and ₁ = 1).

Suppose that in the considered example of a specific example of the application of the proposed method, the length of the code blocks is n = 7, the characters contain three bits (m = 3), the number of elements of the Galois field is eight ( a ₀ ÷ a ₇ ). The common polynomial (polynomial generator) is formed by the polynomial g (X) = a ₁ X + a ₆ . At the same time, the initial expressions for the first and second in a pair of selected code blocks are given the form, respectively:

and

и

- информационные (различающиеся) части первого и второго кодовых слов.Where

and

- informational (different) parts of the first and second code words.

Higher degrees of polynomials are still the same and operations 5 and 6 leave them unchanged. In operation 7, both code blocks are normalized so that the coefficient for the highest power becomes equal to unity, i.e. all the coefficients of the first polynomial are divided by a ₃ and the second polynomial by a ₅ . Then (given that a ₁ = 1):

The polynomials in the current state are not equal to one another, so operation 8 leaves out.

Using operation 9, we find the differential expression:

After normalization (dividing by the coefficient a ₂ ) it becomes equal to:

As a result of the operations of this step, the member of the polynomial of the first code block with the highest power of six disappeared, the number of members in the polynomial decreased by one. After this, a sequence of similar operations is repeated, as a result, the polynomial of the second code block is reduced. To do this, temporarily multiply the polynomial y ₁ ⁽¹⁾ (X) by I and subtract from the polynomial y ₂ ⁽⁰⁾ (X):

After normalization:

Now, in the second code block, the polynomial member with the highest power of six has disappeared, the number of terms in the polynomial has also decreased by one. Further, with the obtained polynomials, we perform similar operations, as a result of which we obtain even more shortened polynomials of the form (without normalization with respect to the coefficient of the highest power):

After normalization:

The senior degree again decreased by one, the number of terms in the polynomials also decreased. Continuing similar actions, we get:

After normalization:

. После нормировки

.After subtracting them, the degree of the polynomial decreased immediately by two units (due to the resulting combination of coefficients), i.e .:

. After normalization

.

As a result, to form the second polynomial when subtracting y _R4, it must be multiplied not by X, but by X ² , i.e .:

After normalization:

, но теперь на величину X):

. Проводя нормировку, получаем выражение

. Операцией 8 определяется, что полиномы стали равны. Полином стал равным оставшемуся первому полиному

, т.е. выражение для общего полинома g(X) выводится к операции 3 и запоминается.Next, we obtain the difference (repeatedly multiplying the polynomial

, but now on the value of X):

. By normalizing, we get the expression

. Operation 8 determines that the polynomials are equal. Polynom became equal to the remaining polynom

i.e. the expression for the general polynomial g (X) is output to operation 3 and is remembered.

After analyzing all the selected pairs of blocks by operation 4, the polynomial that was filled the most times is displayed. Since the polynomial generator is present in all code blocks, it will occur the maximum number of times and will be output as a result of the diagnostic procedure. The information parts of different blocks are different, and the appearance of the same factors in the information part is unlikely, it is much more likely that the common part will consist only of the generating polynomial.

The probability of occurrence in each analysis cycle of only the generating polynomial g (X) is much higher, and it will be determined the maximum number of times, and all other results appearing randomly will be recorded fewer times. Therefore, operation 4 determines exactly the polynomial g (X), which is the desired result of the diagnosis. For each pair of code words, a gradual cyclic reduction of the order of the corresponding polynomials is performed until only a common polynomial for each pair remains in the pair. If the maximum degree of differing parts is equal to k, then in the general case the general polynomial will receive a maximum in k repetitions of the described set of operations.

Thus, the inventive method allows you to determine the structure of the used encoder without prior information about it and to ensure the effective operation of the decoder for error correction, thereby increasing the noise immunity of information transmission.

Claims (1)

A method for diagnosing non-binary block codes, which includes memorizing N code blocks, extracting pairs of blocks, pairwise comparison and addition of blocks, a bitwise shift of the polynomial, memorizing the type of polynomial, and choosing the maximum number of results, characterized by the comparison of polynomial degrees, division by the coefficients in the highest degree of a polynomial, the determination of the equality of polynomials and the addition of polynomials in the Galois field, in this case, first memorize N code blocks, then remove pairs of code blocks, then the code blocks they are compared and folded in pairs, then the type of the polynomials obtained as a result of the comparison and addition is memorized, and then the maximum number of identical results of the comparison and addition obtained is chosen, and when pairwise comparing and adding code blocks, the maximum powers of the members of the selected polynomials that describe the compared code blocks are compared, shift the smaller polynomial bitwise, then divide all the coefficients of each polynomial by the coefficient with its member of the maximum degree, then compare the larger a line with a shifted lower polynomial, in the case of their equality, the form of the smaller polynomial is stored before the shift;

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