RU2294529C2 - Device for correcting errors in polynomial system of residue classes with usage of pseudo-orthogonal polynomials - Google Patents

Abstract

FIELD: computer engineering, possible use for controlling and correcting error occurring during transfer of information and for performing arithmetic operations in expanded Galois fields GF (2^v).

SUBSTANCE: device contains memory block, adder, error syndrome computation block represented by two-layered neuron network, containing neurons forming first layer and second layer of neurons.

EFFECT: increased speed of detection and correction of errors.

2 dwg, 6 tbl

Description

The device relates to the field of computer technology, in particular, it can be used both to control and correct errors in the transmission of information, and during arithmetic operations in the extended Galois fields GF (2 ^v ).

A device for detecting and correcting errors (USSR No. 714399, 02/08/1980, G 06 F 11/08), containing a register, two blocks of modular convolution, designed to calculate the remainder of the number on the control bases, two modular adders, designed to calculate the error syndrome on the first and second control grounds, a memory unit for storing constants, a third adder designed to obtain a corrected number by summing the erroneous number with the error constant.

The disadvantage of this device is the low speed due to the separate sequential use of modular convolution blocks designed to calculate the remainder of the number on the control bases and adders, to calculate the error syndromes on the first and second control bases.

The technical result achieved by the implementation of the claimed invention is to increase the speed of detection and correction of errors.

The indicated technical result is achieved through the use of a polynomial system of residue classes (PSKV), in which the minimum polynomials p _i (z), i = 1, 2, ..., k + r defined in the extended Galois fields GF are used as the base of the system (2 ^v ) and neural network technologies, as well as the use of pseudo-orthogonal polynomials defined in this PSKV.

The peculiarity of PSKV is that the independence of information processing on the basis of PSKV allows not only to increase the speed and accuracy of processing, but also to ensure the detection and correction of errors during the operation of a computing device of the residue class. If restrictions are imposed on the range of possible changes in the encoded set of polynomials, that is, choose k from n bases of PSCW (k <n), this will allow us to split the full range P of the _complete (z) extended Galois field GF (p ^v ) into two disjoint subsets.

The first subset is called the working range and is determined by the expression

A polynomial A (z) with coefficients from the field GF (p) will be considered allowed if and only if it is an element of the zero interval of the full range P _full (z), that is, it belongs to the working range A (z) ∈P _slave ( z).

The second subset of GF (p ^v ) defined by the product r = nk of control bases

sets the combination of prohibited combinations. If A (z) is an element of the second subset, then this combination is considered to contain an error. Thus, the location of the polynomial A (z) with respect to these two subsets allows us to unambiguously determine whether the code combination A (z) = (α ₁ (z), α ₂ (z), ..., α _{k + r} (z) ) resolved or contains erroneous characters.

The mathematical model for determining the positional characteristic, the normalized trace of the polynomial A (z), is based on the Chinese remainder theorem, according to which

и

- ортогональный базис i-го основания; m_i(z) - вес ортогонального базиса.Where

and

- orthogonal basis of the i-th base; m _i (z) is the weight of the orthogonal basis.

, у которых все остатки равны нулю за исключением основания p_i(z)On the other hand, A (z) can be represented as the sum of orthogonal polynomials

for which all residues are equal to zero except for the base p _i (z)

Then, on the basis of equalities (3) and (4), it is obvious that

If we put the condition, A (z) ∈ Р _slave (z), then

According to the Chinese remainder theorem, a polynomial can be represented as

Then each term of expression (6) represents

- ортогональный базис безизбыточной системы оснований ПСКВ.Where

- the orthogonal basis of the redundant base system PSKV.

в модулярных кодах нашли широкое применение псевдоортогональные полиномы, которые представляют собой ортогональные полиномы, у которых нарушена ортогональность по нескольким основаниямAlong with orthogonal polynomials

pseudo-orthogonal polynomials, which are orthogonal polynomials in which orthogonality is violated for several reasons, are widely used in modular codes

.It is known that if in pseudo-orthogonal polynomials the orthogonality on the control bases is violated, then these polynomials are orthogonal polynomials of the redundant base system of the polynomial residue class system

.

в видеTo obtain pseudo-orthogonal polynomials, we extend the base system p ₁ (z), ..., p _k (z) to r control bases p _{k + 1} (z), ..., p _{k + r} (z) and represent the orthogonal polynomials

as

Expression (8) determines the values of pseudo-orthogonal polynomials in which orthogonality is violated for control reasons.

Substituting expression (8) into equality (6), and taking into account that in the process of performing the operation there is no way beyond P _slave (z), we obtain

Therefore fair

согласноThus, on the basis of expression (9) and using the values of pseudo-orthogonal polynomials defined by equality (8), we can calculate the values of the residues from the control bases

according to

и значений α_k+1(z), ..., α_k+r(z), поступающих на вход устройства коррекции ошибок, можно определить синдром ошибки согласно выражениюThen based on the obtained values

and the values α _{k + 1} (z), ..., α _{k + r} (z) received at the input of the error correction device, it is possible to determine the error syndrome according to the expression

If the error syndrome is zero, i.e.

then the original polynomial A (z) ∈P _slave (z) is allowed and does not contain errors. Otherwise, the modular combination is prohibited. Then, depending on the magnitude of the error syndrome, error correction is carried out, i.e.

где (0, ..., Δα(z), ..., 0) - вектор ошибки модулярного кода; Δα_i(z) - глубина ошибки по i-му модулю;

.

where (0, ..., Δα (z), ..., 0) is the error vector of the modular code; Δα _i (z) is the error depth in the i-th module;

.

The structure of the device is presented in figure 1. The device consists of register 2, intended for storing residuals on the working and control base of the control switch for the detection of errors, the input of which is connected to the input 1 of the device, the unit for calculating the error syndrome 3, the inputs of which are connected to the outputs of register 2 , memory blocks 4, designed to store constants - error vectors, the inputs of which are connected to the outputs of the error syndrome calculation unit 3, adder 5, which corrects errors by summing and corrected (error) combination of the UCS code with the error vector, the first, second and third inputs of which are connected respectively to the first three outputs of register 2 through which the remains of the working bases are transmitted, and the fourth input is connected to the output of memory unit 4, the output of adder 5 is output 6 of the device .

The operation of the device for error correction in a polynomial system of residue classes using pseudo-orthogonal polynomials is as follows. At the input 1 of the device for error correction, a controlled number in polynomial form

where α _i (z) is the remainder of the polynomial A (z) modulo p _i (z); p ₁ (z), p ₂ (z), ..., p _k (z) are the working bases of the PSKV system; p _{k + 1} (z), p _{k + 2} (z) - control grounds. This code is written to register 2. Then, from the output of register 2, the PSCW code in binary form is fed to the input of the error syndrome calculation unit 3, which is a two-layer neural network. The modular PSKV code A (z) = (α ₁ (z), α ₂ (z), ..., α _k (z), α _{k + 1} (z), α _{k + 2} (z)) is written by neurons of the first layer of the unit for calculating the error syndrome 3. From the output of the neurons of the first layer, the signals are sent in binary form to the second layer of the neural network, which performs the calculation of the error syndrome according to expression (11). From the output of the neurons of the second layer, the value of the error syndrome in the binary code θ _{k + 1} (z), ..., θ _{k + r} (z) is fed to the input of the memory unit 4, where the corresponding error constant is selected from there. This error constant is fed to the fourth input of adder 5, where it is summed with the value A (z) = (α ₁ (z), α ₂ (z), ..., α _k (z)), presented on working grounds, received from the outputs of register 3 to the first three inputs of the adder 5. The corrected value A (z) according to equality (12) from the output of the adder 5 is fed to the output 6 of the device.

As an example, consider the extended Galois field GF (2 ⁴ ), in which the following bases are defined:

p ₁ (z) = z + 1; p ₂ (z) = z ² + z + l; p ₃ (z) = z ⁴ + z ³ + z ² + z + 1; p ₄ (z) = z ⁴ + z ³ +1; p ₅ (z) = z ⁴ + z + 1, where p ₁ (z), p ₂ (z), p ₃ (z) are the working bases, p ₄ (z), p ₅ (z) are the control bases of PSCW . Then, according to expression (1), we have P _slave (z) = z ⁷ + z ⁶ + z ⁵ + z ² + z + 1.

The following orthogonal bases are defined in the complete base system of the PSKV field GF (2 ⁴ ):

Using the comparability of the orthogonal bases of the redundant and complete systems in the operating range, we determine the values of the orthogonal bases of the redundant base system p ₁ (z) = z + 1; p ₂ (z) = z ² + z + 1; p ₃ (z) = z ⁴ + z ³ + z ² + z + 1. Then

Based on the obtained values, we define all pseudo-orthogonal polynomials for the PSKV GF (2 ⁴ ), taking into account the impossibility of going beyond the working range P _slave (z) = z ⁷ + z ⁶ + z ⁵ + z ² + z + 1. The obtained values are presented in table 1.

Table 1
Pseudo-orthogonal polynomials of PSCW fields GF (2 ⁴ ) The foundation of PSKV Psedo-orthogonal polynomial p ₁ (z) = z + 1 (1, 0, 0, z ³ + z + 1, z) p ₂ (z) = z ² + z + 1 (0, 1, 0, z ² + z + 1, z ³ +1) (0, z, 0, z ³ + z, z ² + z + 1) (0, z + 1, 0, z ³ + z ² +1, z ³ + z ² + z) p ₃ (z) = z ⁴ + z ² + z ² + z + 1 (0, 0, 1, z ³ + z ² +1, z ³ + z) (0, 0, z, z + 1, z ² + z + 1) (0, 0, z + 0, z ³ + z ² + z, z ³ + z ² +1) (0, 0, z ² , z, z ³ ) (0, 0, z ² + l, z ³ + z ² + z + 1, z) (0, 0, z ² + z, 1, z ³ + z ² + z + 1) (0, 0, z ² + z + 1, z ³ + z ² , z ² +1) (0, 0, z ³ , z ² , z + l) (0, 0, z ³ +1, z ³ +1, z ³ +1) (0, 0, z ³ + z, z ² + z + 1, z ² ) (0, 0, z ³ + z + 1, z ³ + z, z ³ + z ² + z) (0, 0, z ² + z ² , z ² + z, z ³ + z + 1) (0, 0, z ³ + z ² +1, z ³ + z + 1, 1) (0, 0, z ³ + z ² + z, z ² +1, z ³ + z ² ) (0, 0, z ³ + z ² + z + 1, z ³ , z ² + z)

If the polynomial A (z) ∈P is _slave , then

where the left side of the equality is presented on the working grounds p ₁ (z), p ₂ (z), p ₃ (z), the middle part of the equality is obtained on the basis of expression (10), and the right side is the modular combination of the base system p ₁ (z) , p ₂ (z), p ₃ (z), p ₄ (z), p ₅ (z) of the extended Galois field GF (2 ⁴ ), received at the input of the device for error correction in a polynomial system of residue classes using pseudo-orthogonal polynomials. Then according to (11) we have

Otherwise

The result obtained indicates that the modular combination A (z) received at the input of the device contains an error that must be corrected. Table 2 presents the values of the error vector (0, ..., Δα _i (z), ..., 0) - the modular code for various values of the error syndrome for the PSCW field GF (2 ⁴ ).

table 2
Values of the error vector of the modular code of the field GF (2 ⁴ ) The foundation of PSKV Error Vector Value θ ₄ (z) θ ₅ (z) 0 0 (0, 0, 0) z ³ + z + 1 z (one hundred) z ² + z + 1 z ³ +1 (0, 1, 0) z ³ + z z ² + z + 1 (0, z, 0) z ³ + z ² +1 z ³ + z ² + z (0, z + 1, 0) z ³ + z ² +1 z ³ + z (0, 0, 1) z + 1 z ² + z + 1 (0, 0, z,) z ³ + z ² + z z ³ + z ² + l (0, 0, z + 1) z z ³ (0, 0, z ² ) z ³ + z ² + z + l z (0, 0, z ² +1) one z ³ + z ² + z + 1 (0, 0, z ² + z) z ³ + z ² z ² +1 (0, 0, z ² + z + 1) z ² z + 1 (0, 0, z ³ ) z ³ +1 z ³ + l (0, 0, z ³ +1) z ² + z + 1 z ² (0, 0, z ³ + z) z ³ + z z ³ + z ² + z (0, 0, z ³ + z + 1) z ² + z z ³ + z + 1 (0, 0, z ³ + z ² ) z ³ + z + 1 one (0, 0, z ³ + z ² +1) z ² +1 z ³ + z ² (0, 0, z ³ + z ² + z) z ³ z ² + z (0, 0, z ³ + z ² + z + l)

Let the polynomial A (z) = z ⁵ + z ⁴ + z be supplied to the input 1 of the device, the value of which in the modular code is represented by A (z) = (1, z + 1, z ³ + z ² , 0, z ² + z ⁴ +1).

Then, according to expression (8) and the data presented in table 1, we obtain the following pseudo-orthogonal polynomials A _i (z):

According to the expression (10) we determine the values of the residues on the control grounds:

According to the expression (11) we determine the error syndrome:

From memory block 5, in accordance with θ ₄ (z) = 0, θ ₅ (z) = 0, the quantity (0, 0, 0) is selected, which is added to A (z) = (1, z + 1, z ³ + z ² ), in the adder 6 with the formation of the output

Suppose that an error occurred on the third base in the adopted combination and its depth is Δα ₃ (z) = z ² + z. Then we have

Then, according to expression (8) and the data presented in table 1, we obtain the following pseudo-orthogonal polynomials A _i (z):

According to the expression (10) we determine the values of the residues on the control grounds:

According to the expression (11) we determine the error syndrome:

From memory block 5, in accordance with θ ₄ (z) = 1, θ ₅ (z) = z ³ + z ² + z + 1, the quantity (0, 0, z ² + z) is selected, which is added to A (z) = (1, z + 1, z ³ + z) in adder 6 with output formation

A bug in the modular code has been fixed.

The error syndrome calculation unit is shown in FIG. The error syndrome calculation unit has 5 inputs, the first three of which are the residues α ₁ (z), α ₂ (2), α ₃ (z) based on the working base p ₁ (z), p ₂ (z), p ₃ ( z), in the fourth and fifth, residues α ₄ (z) and α ₅ (z) are received for the control bases p ₄ (z) and p ₅ (z). The syndrome calculation unit is a two-layer neural network. The first layer contains 15 neurons. The inputs of neurons 7, 8-9, 10-13 are connected respectively to the first, second and third input of the error syndrome calculation unit. The inputs of neurons 14-17 and 18-21 are connected respectively to the fourth and fifth inputs of the error syndrome calculation unit. At the inputs of neurons 7, 8-9, 10-13 in binary form, the residues α ₁ (z), α ₂ (z), α ₃ (z) are received along the three working bases of the PSCW. The binary code α ₄ (z) is supplied to neurons 14-17 by the first control module p ₄ (z) = z ⁴ + z ³ +1. On neurons 18-21 in the binary code, α ₅ (z) enters the binary code using the second control module p ₅ (z) = z ⁴ + z + 1. The highest bits of the residues of the polynomial α ₁ (z), α ₂ (z), α ₃ (z), α ₄ (z), α ₅ (z) are recorded respectively in the neurons 7, 8, 10, 14, 18 of the first layer. The second layer of the neural network contains 8 neurons that perform the basic operation according to expression (11), the first four neurons 22-25 determine the value of θ ₄ (z), the remaining neurons 26-29 determine the value of θ ₅ (z). The inputs of the neuron 22 of the second layer are connected to the outputs of the neurons 7, 8, 13, 14 of the first layer. The inputs of the neuron 23 of the second layer are connected to the outputs of the neurons 9, 10, 13, 15 of the first layer. The inputs of the neuron 24 of the second layer are connected to the outputs 7, 8, 9, 11, 12, 16 of the neurons of the first layer. The inputs of the neuron 25 of the second layer are connected to the outputs 7, 9, 12, 13, 17 of the neurons of the first layer. The inputs of the neuron 26 of the second layer are connected to the outputs 9, 11, 13, 18 of the neurons of the first layer. The inputs of the neuron 27 of the second layer are connected to the outputs 8, 12, 19 of the neurons of the first layer. The inputs of the neuron 28 of the second layer are connected to the outputs 7, 8, 10, 12, 13, 20 of the neurons of the first layer. The inputs of the neuron 29 of the second layer is connected to the outputs of the neurons 8, 9, 10, 12, 21 neurons of the first layer. The highest values of the error syndrome θ ₄ (z) and θ ₅ (z) are respectively calculated in neurons 22 and 26. The outputs of the neurons of the second layer are the output of the error syndrome calculation unit.

Consider the process of the unit for computing the error syndrome using examples. Let the modular code A (z) = (1, z + 1, z ³ + z ² , 0, z ² + z + 1) be fed to the input of the error correction device. This code is fed to the inputs of the neurons of the first layer of the error syndrome calculation unit. The signals at the output of neurons of the first layer are presented in table 3.

Table 3
Signals at the output of neurons of the first layer Neurons 7 8 9 10 eleven 12 13 fourteen fifteen 16 17 eighteen 19 twenty 21 Signal output one one one one one 0 0 0 0 0 0 0 one one one

The signals from the outputs of the neurons of the first layer are fed to the corresponding inputs of the neurons of the second layer. Table 4 shows the values of the signals at the input and output of the neurons of the second layer. The symbol “-” corresponds to the absence of communication between the neurons of the second and first layer.

Table 4
Signals at the output of neurons of the second layer Neurons 7 8 9 10 eleven 12 13 fourteen fifteen 16 17 eighteen 19 twenty 21 exit 22 one - one - - - 0 0 - - - - - - - 0 23 - - one one - - 0 - 0 - - - - - - 0 24 one one one - one 0 - - - 0 - - - - - 0 25 one - one - - 0 0 - - - 0 - - - - 0 26 - - one - one - 0 - - - - 0 - - - 0 27 - one - - - 0 - - - - - - one - - 0 28 one one - one - 0 0 - - - - - - one - 0 29th - one one one - 0 - - - - - - - - one 0

Suppose that an error occurred on the third base in the adopted combination and its depth is Δα ₃ (z) = z ² + z. Then we have

This code is fed to the inputs of the neurons of the first layer of the error syndrome calculation unit. The signals at the output of neurons of the first layer are presented in table 5.

Table 5
Signals at the output of neurons of the first layer Neurons 7 8 9 10 eleven 12 13 fourteen fifteen 16 17 eighteen 19 twenty 21 Signal output one one one one 0 one 0 0 0 0 0 0 one one one

The signals from the outputs of the neurons of the first layer are fed to the corresponding inputs of the neurons of the second layer. Table 6 presents the values of the signals at the input and output of the neurons of the second layer.

Table 6
Signals at the output of neurons of the second layer Neurons 7 8 9 10 eleven 12 13 fourteen fifteen 16 17 eighteen 19 twenty 21 exit 22 one - - - - 0 0 - - - - - - - 0 23 - - one - - 0 - 0 - - - - - - 0 24 one one - 0 one - - - 0 - - - - - 0 25 one - - - one 0 - - - 0 - - - - one 26 - - - 0 - 0 - - - - 0 - - - one 27 - one - - - one - - - - - - one - - one 28 one one - one - one 0 - - - - - - one - one 29th - one one one - one - - - - - - - - one one

Thus, at the output of the error syndrome calculation unit, the values θ ₄ (z) = 1, θ ₅ (z) = z ³ + z ² + z + 1 are determined. The data obtained coincide with the control miscalculations presented earlier.

Claims (1)

A device for error correction in a polynomial system of residue classes using pseudo-orthogonal polynomials contains a device input, a memory unit for storing weight coefficients, and the memory unit is connected to the fourth input of the adder, the output of which is the output of the device, characterized in that it additionally an error syndrome calculation unit has been introduced, the inputs of which are connected to the device inputs, and the outputs are connected to the inputs of the memory unit and to the first three inputs of the adder, while the error syndrome calculation unit is a two-layer neural network, the first layer of which contains fifteen neurons, the second layer of the neural network contains eight neurons, while the first layer of the neural network is used to record a controlled number represented in the polynomial residue class system, and the second layer is used to calculate error syndrome, and the inputs of the neurons of the said first layer are the input of the unit for calculating the error syndrome, the outputs of the neurons of the said second layer are the output and a unit for calculating the error syndrome, while the inputs of neurons from the first to the seventh mentioned first layer receive balances on the working bases of the mentioned controlled number, the inputs of neurons from the eighth to fifteenth of the mentioned first layer are left over the control bases of the mentioned controlled number, inputs of the first neuron of the mentioned the second layer is connected to the outputs of the first, second, seventh and eighth neurons of the first layer, the inputs of the second neuron of the second layer are connected to the output the third, fourth, seventh and ninth neurons of said first layer, the inputs of the third neuron of said second layer are connected to the outputs of the first, second, third fifth, sixth and tenth neurons of said first layer, the inputs of the fourth neuron of said second layer are connected to the outputs of the first, third, of the sixth, seventh and eleventh neurons of said first layer, the inputs of the fourth neuron of said second layer are connected to the outputs of the third, fifth, seventh and twelfth neurons of said first layer, the inputs of the fifth neuron of the said second layer are connected to the outputs of the second, sixth and thirteenth neurons of the said first layer, the inputs of the seventh neuron of the said second layer are connected to the outputs of the first, second, fourth, sixth, seventh and fourteenth neurons of the said first layer the second layer is connected to the outputs of the second, third, fourth, sixth and fifteenth neurons of said first layer.

Images