RU2390051C2 - Device for spectral error detection and correction in codes of polynomial system of residue classes - Google Patents

Abstract

FIELD: information technology.

SUBSTANCE: invention relates to modular neurocomputer apparatus and is designed for detecting errors in code structures of the position-independent code of a polynomial system of residue classes (PSRC) presented in augmented Galois fields GF(2^V). The device has a register, an interval polynomial calculation unit, a correcting adder, a spectral analysis unit which is a four-layer neural network. The first layer is designed for recording the interval polynomial which is presented in form of a binary code. The second layer is designed for calculating the first spectral components from control bases. The third layer is designed for inverting the obtained values. The fourth layer is designed fro calculating the correction value which is presented in a polynomial system of residue classes.

EFFECT: reduced hardware expenses.

2 dwg, 10 tbl

Description

The device relates to computer technology and can be used both for monitoring and correcting errors in the transmission of information, and during arithmetic operations in computers.

A device is known for spectral detection and correction of errors in codes of a polynomial residue class system (DPSC) (patent RU 2301441, class G06F 7/72), which contains three information inputs and two control inputs, a register, an interval polynomial calculation unit, a spectral analysis unit, a constant storage device, correcting adder, output.

The disadvantages of the known device are its complexity and significant circuit costs.

The technical result achieved by the implementation of the claimed invention is to reduce hardware costs for determining the location and depth of error in the modular code PSCW and its subsequent correction.

Said technical result is achieved due to removal of permanent memory structure, wherein the correction values stored Δα ^armature (z), and the introduction of the block of spectral analysis, representing a two layer neural network (NN) of the third and fourth layers of neurons.

Functional diagram of the device shown in figure 1. The device contains an input 1, which serves the polynomial code

where α _i (z) - the remains on the working and control grounds PSKV p _i (z);

i = 1 ÷ n - the number of bases PSKV,

it is connected to the input of register 2, the output of which is connected to the input of the unit for calculating the interval polynomial 3, which implements the calculation in the time domain

where Q _i (z) is the quotient of dividing the orthogonal basis B _i (z) by

k - the number of working bases PSKV;

r = n-k is the number of control bases;

B _m ^* (z) is the orthogonal basis of the lossless PSKV.

The output of block 3 is connected to the input of the spectral analysis block 4, the output of which is connected to the second input of the correcting adder 5, the first input of which is connected to the output of register 2, through which the remnants of the PSCW modular code are transmitted on working and control grounds. The output of the correcting adder 5 is the output 6 of the device.

Consider the operation of the device. At the input 1 of the device is fed three residues on working grounds p ₁ (z) = z + 1; p ₂ (z) = z ² + z + 1; p ₃ (z) = z ⁴ + z ³ + z ² + z + 1 and two control bases p ₄ (z) = z ⁴ + z ³ +1 and p ₅ (z) = z ⁴ + z + 1. In this case, the working range will be equal to

It is known that if the polynomial A (z), presented in the form of PSCW, belongs to the working range, then it does not contain errors.

Otherwise, A (z) = (α ₁ (z), α ₂ (z), α ₃ (z), α ₄ (z), α ₅ (z)) contains an error.

Moreover, if the modular code A (z) does not contain errors, then certain spectral components of the interval polynomial (2) defined by the expression

where S _j (Z) is the jth spectral component of the polynomial s (z);

β _{k + l} is the antiderivative element of the extended field GF (p ^v ) generated by the polynomial p _{k + l} (z); l = 1, 2, ..., r, j = 0, ..., p ^v-1 -1,

will be equal to 0.

Such spectral components will be S ₁ (Z ₁ ), S ₂ (Z ₁ ), S ₄ (Z ₁ ), S ₈ (Z ₁ ), as well as S ₇ (Z ₁ ), S ₁₁ (Z ₁ ), S ₁₃ (Z ₁ ), S ₁₄ (Z ₁ ). These spectral components of the interval polynomial s (z) correspond to the roots of the control bases p ₄ (z) and p ₅ (z), respectively.

The appearance of an error in the n-dimensional combination A (z) on the ith base and depth Δα _i (z) leads to a shift of the latter in the time domain by

those.

In this case, a shift in the spectrum is observed. Table 1 shows the spectral shifts for various errors for all PSCW bases of the extended Galois field GF (2 ⁴ ).

Table 1 Alignment offsets in GF (2 ⁴ ) when errors occur in PSCW Base Error Depth Δα _i Polynomial s (z) The offset value S ₁ (Z) p ₄ (z) = z ⁴ + z ³ +1 p ⁵ (z) = z ⁴ + z + 1 p ₁ (z) = z + 1 one z ⁷ + z ⁴ + z ² + z β ³ β ¹¹ p ₂ (z) = z ² + z + 1 one z ⁷ + z ⁵ + z ² + z + 1 β ⁵ β ⁹ z z ⁸ + z ⁶ + z ³ + z ² + z + 1 β ⁸ β ⁵ p ₃ (z) = z ⁴ + z ³ + z ² + z + 1 one z ⁷ + z ⁴ + z ³ + z + 1 β ⁹ β ⁴ z z ⁸ + z ⁵ + z ⁴ + z ² + z β ¹⁰ β ⁵ z ² z ⁹ + z ⁶ + z ⁵ + z ³ + z ² +1 β ¹⁴ β ¹³ z ³ z ¹⁰ + z ⁷ + z ⁶ + z ⁴ + z ³ + z one β ¹⁴ p ₄ (z) = z ⁴ + z ³ +1 one z ⁷ + z ⁴ + z ³ β ¹³ 0 z z ⁸ + z ⁵ + z ⁴ β ¹⁴ 0 z ² z ⁹ + z ⁶ + z ⁵ one 0 z ³ z ¹⁰ + z ⁷ + z ⁶ β 0 p ₅ (z) = z ⁴ + z + 1 one z ⁵ + z ⁴ + z 0 β ¹⁰ z z ⁶ + z ⁵ + z ² 0 β ¹¹ z ² z ⁷ + z ⁶ + z ³ 0 β ¹² z ³ z ⁸ + z ⁷ + z ⁴ 0 β ¹³

Analyzing the displacements of the first harmonic S ₁ (Z), it is difficult to make a correspondence between its value and the location of the error Δα _i (z), which is shown in Table 2.

The analysis of the table shows that it is possible to determine the values of Δα _i (z) using the third layer of neurons of the spectral analysis unit.

The structure of the block of spectral analysis is shown in figure 2. It is a four-layer neural network, the first layer of which contains fifteen neurons 7-21, the second layer - eight neurons 22-29, the third layer - eight neurons 30-37, and the fourth layer - fifteen neurons 38-52. The neurons of the first layer distribute the values received at their inputs, the neurons of the second layer implement the basic summing operation modulo two, the neurons of the third layer implement the basic operation “NOT”, and the neurons of the fourth layer implement the “logical AND” operation. Moreover, the inputs of the neuron 22 of the second layer are connected to the outputs 7, 11, 12, 13, 14, 16, 18, 19 of the neurons of the first layer. The inputs of the neuron 23 of the second layer are connected to the outputs 8, 12, 13, 14, 15, 17, 19, 20 of the neurons of the first layer. The inputs of the neuron 24 of the second layer are connected to the outputs 9, 13, 14, 15, 16, 18, 20, 21 of the neurons of the first layer. The inputs of the neuron 25 of the second layer are connected to the outputs 10, 11, 12, 13, 15, 17, 18, 21 of the neurons of the first layer. The outputs of neurons 22, 23, 24, 25 are respectively the zero, first, second and third bits of the first spectral component S ₁ (Z ₁ ) modulo p ₄ (z) = z ⁴ + z ³ +1.

The inputs of the neuron 26 of the second layer are connected to the outputs 7, 11, 14, 15, 16, 18, 19, 20 of the neurons of the first layer. The inputs of the neuron 27 of the second layer are connected to the outputs 8, 11, 12, 14, 16, 17, 18, 19 of the neurons of the first layer. The inputs of the neuron 28 of the second layer are connected to the outputs 9, 12, 13, 15, 17, 18, 19, 20 of the neurons of the first layer. The inputs of the neuron 29 of the second layer are connected to the outputs 10, 13, 14, 16, 18, 19, 20, 21 of the neurons of the first layer. The outputs of neurons 26, 27, 28, 29 are respectively the zero, first, second and third bits of the first component S ₁ (Z ₁ ) modulo p ₅ (z) = z ⁴ + z + 1.

The input of the neuron 30 of the third layer is connected to the output of the neuron 22 of the second layer. The input of the neuron 31 of the third layer is connected to the output of the neuron 23 of the second layer. The input of the neuron 32 of the third layer is connected to the output of the neuron 24 of the second layer. The input of the neuron 33 of the third layer is connected to the output of the neuron 25 of the second layer. The input of the neuron 34 of the third layer is connected to the output of the neuron 26 of the second layer. The input of the neuron 35 of the third layer is connected to the output of the neuron 27 of the second layer. The input of the neuron 36 of the third layer is connected to the output of the neuron 28 of the second layer. The input of the neuron 37 of the third layer is connected to the output of the neuron 29 of the second layer. Neurons 30-37 of the third layer implement the operation "logical NOT".

The inputs of the neuron 38 of the fourth layer are connected to the outputs 25, 27, 28, 29 of the neurons of the second layer and the outputs 30, 31, 32, 34 of the neurons of the third layer. The inputs of the fourth layer neuron 39 are connected to the outputs of the second layer neurons 22, 23, 25, 27, 29 and the third layer neurons 32, 34, 36. The inputs of the neuron 40 of the fourth layer are connected to the outputs 23, 24, 25, 27, 28 of the neurons of the second layer and the outputs 30, 34, 37 of the neurons of the third layer. The inputs of the neuron 41 of the fourth layer are connected to the outputs 22, 24, 26, 27 of the neurons of the second layer and the outputs 31, 33, 36, 37 of the neurons of the third layer. The inputs of the fourth layer neuron 42 are connected to the outputs of the second layer neurons 23, 25, 27, 28 and the outputs of the third layer neurons 30, 32, 34, 37. The inputs of the neuron 43 of the fourth layer are connected to the outputs 24, 25, 26, 28, 29 of the neurons of the second layer and the outputs 30, 31, 35 of the neurons of the third layer. The inputs of the fourth layer neuron 44 are connected to the outputs 22, 26, 29 of the second layer neurons and the outputs 31, 32, 33, 35, 36 of the third layer neurons. The inputs of the fourth layer neuron 45 are connected to the outputs 23, 24 of the second layer neurons and the outputs 30, 33, 34, 35, 36, 37 of the third layer neurons. The inputs of the fourth layer neuron 46 are connected to the outputs 24, 25 of the second layer neurons and the outputs 30, 31, 34, 35, 36, 37 of the third layer neurons. The inputs of the fourth layer neuron 47 are connected to the outputs of the 22 neurons of the second layer and the outputs 31, 32, 33, 34, 35, 36, 37 of the neurons of the third layer. The inputs of neuron 48 of the fourth layer are connected to the outputs of 23 neurons of the second layer and outputs 30, 32, 33, 34, 35, 36, 37 of the neurons of the third layer. The inputs of the fourth layer neuron 49 are connected to the outputs 26, 27, 28 of the second layer neurons and the outputs 30, 31, 32, 33, 37 of the third layer neurons. The inputs of the fourth layer neuron 50 are connected to the outputs of the second layer neurons 27, 28, 29 and the outputs 30, 31, 32, 33, 34 of the third layer neurons. The inputs of the neuron 51 of the fourth layer are connected to the outputs 26, 27, 28, 29 of the neurons of the second layer and the outputs 30, 31, 32, 33 of the neurons of the third layer. The inputs of the neuron 52 of the fourth layer are connected to the outputs 26, 28, 29 of the neurons of the second layer and the outputs 30, 31, 32, 33, 35 of the neurons of the third layer.

The neurons 38-52 of the fourth layer implement the basic operation of "logical multiplication". The values Δα _i (z) B _i (z) modP _full (z), i = 1, 2, 3, 4, 5, presented in the modular code, are taken from the outputs of these neurons.

In this case, the correction value Δα ₁ ^cor (z) = Δα ₁ (z) B ₁ (z) modP is _full (z), where

- полный диапазон; снимается с выхода нейрона 38 четвертого слоя. Корректирующее значение Δα₂ ^кор(z)=Δα₂(z)B₂(z)modP_полн(z) снимается с выходов нейронов 39 и 40 четвертого слоя. Корректирующее значение Δα₃ ^кор(z)=Δα₃(z)В₃(z)modР_полн(z) снимается с выходов нейронов 41-44 четвертого слоя. Корректирующее значение Δα₄ ^кор(z)=Δα₄(z)В₄(z)modР_полн(z) снимается с выходов нейронов 45-48 четвертого слоя. Корректирующее значение Δα₅ ^кор(z)=Δα₅(z)В₅(z)modР_полн(z) снимается с выходов нейронов 49-52 четвертого слоя. Данные значения Δα_i ^кор(z)=Δα_i(z)B_i(z)modP_полн(z) представлены в двоичном коде, при этом снимаются с выходов 38, 40, 44, 48, 52 нейронов четвертого слоя. Выходы нейронов четвертого слоя являются выходом блока спектрального анализа.

- full range; removed from the output of neuron 38 of the fourth layer. The corrective value Δα ₂ ^core (z) = Δα ₂ (z) B ₂ (z) modP _full (z) is removed from the outputs of neurons 39 and 40 of the fourth layer. The corrective value Δα ₃ ^cor (z) = Δα ₃ (z) B ₃ (z) modР _complete (z) is removed from the outputs of neurons 41-44 of the fourth layer. The corrective value Δα ₄ ^cor (z) = Δα ₄ (z) In ₄ (z) modР _complete (z) is removed from the outputs of neurons 45-48 of the fourth layer. The corrective value Δα ₅ ^cor (z) = Δα ₅ (z) At ₅ (z) modР _complete (z) is removed from the outputs of neurons 49-52 of the fourth layer. These values of Δα _i ^core (z) = Δα _i (z) B _i (z) modP _full (z) are presented in binary code, while they are removed from the outputs 38, 40, 44, 48, 52 of the fourth layer neurons. The outputs of the fourth layer neurons are the output of the spectral analysis unit.

The device operates as follows. At the input 1 of the device, the modular polynomial code A (z) = (α ₁ (z), α ₂ (z), α ₃ (z), α ₄ (z), α ₅ (z)) is supplied, which is input to the register 2, designed to store the received combination, this modular code has 3 working bases p ₁ (z) = z + 1, p ₂ (z) = z ² + z + 1, p ₃ (z) = z ⁴ + z ³ + z ² + z + 1, which form a range

and two control bases p ₄ (z) = z ⁴ + z ³ +1 and p ₅ (z) = z ⁴ + z + 1. Let as the initial polynomial choose A (z) = z ⁶ + z + 1 belonging to the working range. Then this polynomial is represented in the form of the PSKV code A (z) = (1, z, 1, z ³ + z ² , z ³ + z ² + z + 1).

From the output of register 2, the modular code is fed to the input of the unit for calculating the interval polynomial 3. For the PSKV system, we have orthogonal bases

B ₁ (z) = z ¹⁴ + z ¹³ + z ¹² + z ¹¹ + z ¹⁰ + z ⁹ + z ⁸ + z ⁷ + z ⁶ + z ⁵ + z ⁴ + z ³ + z ² + z + 1;

B ₂ (z) = z ¹⁴ + z ¹³ + z ¹¹ + z ¹⁰ + z ⁸ + z ⁷ + z ⁵ + z ⁴ + z ² + z;

B ₃ (z) = z ¹⁴ + z ¹³ + z ¹² + z ¹¹ + z ⁹ + z ⁸ + z ⁷ + z ⁶ + z ⁴ + z ³ + z ² + z;

B ₄ (z) = z ¹⁴ + z ¹³ + z ¹² + z ¹¹ + z ⁹ + z ⁷ + z ⁶ + z ³ ;

B ₅ (z) = z ¹² + z ⁹ + z ⁸ + z ⁶ + z ⁴ + z ³ + z ² + z.

Then

B ₁ ^* (z) = B ₁ (z) modP _slave (z) = z ⁶ + z ⁴ + z ³ + z ² +1;

B ₂ ^* (z) = B ₂ (z) modP _slave (z) = z ⁶ + z ⁵ + z + 1;

B ₃ ^* (z) = B ₃ (z) modP _slave (z) = z ⁵ + z ⁴ + z ³ + z ² +1.

According to expression (2), we have s ₁ (z) = z ¹⁰ + z ⁹ + z ⁷ + z ⁶ + z ⁵ + z ² + z.

This value of the interval polynomial is fed to the inputs of the neurons of the first layer of the block of spectral analysis 4. Table 3 shows the distribution of signals at the output of the neurons of the first layer.

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 0 0 one one 0 one one one 0 one one 0 0 0 0

These values are fed to the neurons of the second layer, at the output of which a signal is obtained in accordance with the data presented in tables 4, 5.

The obtained zero result at the output of the neurons of the second layer goes to the corresponding inputs of the neurons of the third and fourth layers. Since these neurons implement the basic operation “NOT” and the operation of logical multiplication, then the outputs of the neurons of the fourth layer will have a zero result. This signal from the output of the block of spectral analysis 4 is fed to the second input of the correcting adder 5, the first input of which is supplied from register 2 modular code A (z). Corrective adder 5 implements the operation

A _correction (z) = A (z) + Δα ^cor (z) = (1, z, 1, z ³ + z ² , z ³ + z ² + z + 1) + (0,0,0,0, 0) =

= (1, z, 1, z ³ , + z ² , z ³ + z ² + z + 1).

The resulting value is fed to the output of the device 6.

Assume that the error occurred on the second base p ₂ (z) = z ² + z + 1, and its depth Δα ₂ (z) = z. Then the modular combination has the form

A _osh (z) = (1,0,1, z ³ + z ² , z ³ + z ² + z + 1).

This modular code PSKV through input 1 enters the register 2, the output of which is fed to the block calculation interval polynomial 3.

The binary code of the interval polynomial is output from the output of the block for computing the interval polynomial 3

s (z) = z ¹⁰ + z ⁹ + z ⁸ + z ⁷ + z ⁵ + z + 1.

This value goes to the inputs of the neurons of the first layer of the block of spectral analysis 4. The signals at the output of these neurons are determined by table 6.

Table 6 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 one one 0 0 0 one 0 one one one one 0 0 0 0

These values are fed to the inputs of the neurons of the second layer, at the output of which signals appear in accordance with tables 7 and 8.

Non-zero result of spectral components modulo p ₄ (z) = z ⁴ + z ³ + 1-s ₁ (z) = 1110 = z ³ + z ² + z, modulo p ₅ (z) = z ⁴ + z + 1 -s ₂ (z) = 0110 = z ² + z indicates that the combination A _osh (z) contains an error. These signals are fed to the inputs of neurons of the third and fourth layers.

Table 9 presents the signals at the output of the neurons of the third layer.

Table 10 presents the signals at the output of neurons of the fourth layer.

The result obtained at the output of neurons of the fourth layer corresponds to

Δα ₂ ^cor (z) = (0, z, 0,0,0).

This modular code from the output of the block of spectral analysis 4 is fed to the second input of the correcting adder 5, the first input of which is fed the code PSCW from register 2. The correcting adder 5 implements the operation

A _correction (z) = A (z) + Δα ^cor (z) = (1,0,1, z ³ + z ² , z ³ + z ² + z + 1) + (0, z, 0,0, 0) =

= (1, z, 1, z ³ + z ² , z ³ + z ² + z + 1).

The bug is fixed.

Claims (1)

The device for spectral detection and error correction in codes of a polynomial system of residue classes contains a device input, a register, an interval polynomial calculation unit, a spectral analysis unit, a correcting adder, the output of which is the output of the device, the input of the register connected to the input of the device, and the output of the register connected to the input block calculation interval polynomial, the output of which is connected to the input of the block of spectral analysis, in addition, the output of the register is connected to the first input corrective o adder, characterized in that the output of the spectral analysis unit is connected to the second input of the correcting adder, while the spectral analysis unit is a four-layer neural network, the first layer of which contains fifteen neurons, the second layer - eight neurons, the third layer - eight neurons, the fourth layer - fifteen neurons, while the first layer is designed to record an interval polynomial presented in the form of a binary code, the second layer is designed to calculate the first spectral components x for control reasons, the third layer is for inverting the obtained values, and the fourth layer is for calculating the correction value presented in the polynomial system of residue classes, the inputs of the neurons of the first layer are the input of the spectral analysis unit, and the outputs of the neurons of the fourth layer are the output of the spectral analysis unit while the inputs of the first neuron of the second layer are connected to the outputs of the first, fifth, sixth, seventh, eighth, tenth, twelfth, thirteenth neurons of the first layer, the inputs of the second the second layer yron is connected to the outputs of the second, sixth, seventh, eighth, ninth, eleventh, thirteenth, fourteenth neurons of the first layer, the inputs of the third layer neuron are connected to the outputs of the third, seventh, eighth, ninth, tenth, twelfth, fourteenth, fifteenth neurons of the first layer, the inputs of the fourth neuron of the second layer are connected to the outputs of the fourth, fifth, sixth, seventh, ninth, eleventh, twelfth, fifteenth neurons of the first layer, the inputs of the fifth neuron of the second layer are connected are connected to the outputs of the first, fifth, eighth, ninth, tenth, twelfth, thirteenth, fourteenth neurons of the first layer, the inputs of the sixth neuron of the second layer are connected to the outputs of the second, fifth, sixth, eighth, tenth, eleventh, twelfth, thirteenth neurons of the first layer, inputs the seventh neuron of the second layer is connected to the outputs of the third, sixth, seventh, ninth, eleventh, twelfth, thirteenth, fourteenth neurons of the first layer, the inputs of the eighth neuron of the second layer are connected to the outputs of the fourth the eighth, eighth, tenth, twelfth, thirteenth, fourteenth, fifteenth neurons of the first layer, the input of the i-th neuron of the third layer, i = 1 ... 8, is connected to the output of the i-th neuron of the second layer, the inputs of the first neuron of the fourth layer are connected to the outputs of the fourth of the sixth, seventh, eighth neurons of the second layer and the outputs of the first, second, third, fifth neurons of the third layer, the inputs of the second neuron of the fourth layer are connected to the outputs of the first, second, fourth, sixth, eighth neurons of the second layer and the outputs of the third, fifth, seventh neurons of the third layer, the inputs of the third neuron of the fourth layer are connected to the outputs of the second, third, fourth, sixth, seventh neurons of the second layer and the outputs of the first, fifth, eighth neurons of the third layer, the inputs of the fourth neuron of the fourth layer are connected to the outputs of the first, third, fifth, sixth neurons of the second layer and the outputs of the second, fourth, seventh, eighth neurons of the third layer, the inputs of the fifth neuron of the fourth layer are connected to the outputs of the second, fourth, sixth, seventh neurons of the second layer and the outputs of the first about the third, fifth, eighth neurons of the third layer, the inputs of the sixth neuron of the fourth layer are connected to the outputs of the third, fourth, fifth, seventh, eighth neurons of the second layer and the outputs of the first, second, sixth neurons of the third layer, the inputs of the seventh neuron of the fourth layer are connected to the outputs the first, fifth, eighth neurons of the second layer and the outputs of the second, third, fourth, sixth, seventh neurons of the third layer, the inputs of the eighth neuron of the fourth layer are connected to the outputs of the second, third neurons of the second layer and output the first, fourth, fifth, sixth, seventh, eighth neurons of the third layer, the inputs of the ninth neuron of the fourth layer are connected to the outputs of the third, fourth neurons of the second layer and the outputs of the first, second, fifth, sixth, seventh, eighth neurons of the third layer, the inputs of the tenth neuron of the fourth layer are connected to the output of the first neuron of the second layer and the outputs of the second, third, fourth, fifth, sixth, seventh, eighth neurons of the third layer, the inputs of the eleventh neuron of the fourth layer are connected to the output of the second neuron in of the second layer and the outputs of the first, third, fourth, fifth, sixth, seventh, eighth neurons of the third layer, the inputs of the twelfth neuron of the fourth layer are connected to the outputs of the fifth, sixth, seventh neurons of the second layer and the outputs of the first, second, third, fourth, eighth neurons of the third layer, the inputs of the thirteenth neuron of the fourth layer are connected to the outputs of the sixth, seventh, eighth neurons of the second layer and the outputs of the first, second, third, fourth, fifth neurons of the third layer, the inputs of the fourteenth neuron of the fourth loya are connected to the outputs of the fifth, sixth, seventh, eighth neurons of the second layer and the outputs of the first, second, third, fourth neurons of the third layer, the inputs of the fifteenth neuron of the fourth layer are connected to the outputs of the fifth, seventh, eighth neurons of the second layer and the outputs of the first, second, third , fourth, sixth neurons of the third layer.

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