# Device for transforming a number from polynomial system of residual classes to positional code with error correction

FIELD: computer engineering, possible use in devices for transformation of numbers from polynomial system of residual classes to positional code.

SUBSTANCE: device contains shift register, synchronization block, constant memory block, group of AND elements, positional accumulating adder, error detection block, data storage block, modulus two correcting adder. Error detection block is made in form of three-layered neuron network.

EFFECT: increased speed of transformation, expanded functional capabilities of device due to ensured error correction.

2 cl, 2 dwg, 5 tbl

The invention relates to computer technology and can be used for high-speed transfer of numbers from a polynomial system classes deductions (PSCW) in the positional correction code errors.

A device for converting a number from the system of residual classes (JUICE) in the position code (as the USSR №1005028, G06F 5/02)containing positional accumulating adder whose output is the output device, and a synchronization unit, characterized in that in order to improve performance, it contains the memory block is constant, the group of items, shift register, a group of inputs which is the input device, and a control input connected to the first output of the synchronization unit, the first inputs of elements And groups merged and connected to the output LSB shift register, and the second inputs connected to the corresponding data outputs of the memory block constants, the control input of which is connected to the second output of the synchronization unit, the outputs of the elements And groups connected to the corresponding inputs positional accumulating adder.

The disadvantage of this device is the inability to correct the error.

One of the most promising methods for the detection and correction of errors is to compute senior coefficients of the generalized article system (OPS).

In a slave is the Reference book on digital computing. /Edited Benamrouche. K., Machinery, 1974, 512 S.) on pages 33-35 presents methods for calculating the coefficients of the OPS. These methods are characterized by significant time (calculation time is proportional to the number of bases of the system).

To reduce calculation time senior coefficients OPS is possible through the use of polynomial systems PSCW and pseudoorthogonal bases.

If the original number And be represented in polynomial form, and as a reason to choose the minimal polynomial of an extended Galois fields, this number can be represented in the form:

A(z)=(α_{1}(z), α_{2}(z),... α_{n}(z)),

where α_{i}(z)≡A(z)mod p_{i}(z); p_{i}(z) - minimal polynomial.

If as a business reason to choose k minimal polynomials PSCV (k<n), the data determined operating range

If A(z)∈P_{slave}(z)A(z) - resolved.

Pseudoorthogonal polynomials are orthogonal polynomials that have violated the orthogonality on several grounds

It is known that if pseudoorthogonal polynomials broken orthogonality control on the grounds that these polynomials are orthogonal polynomials nonredundant system grounds Pauline is premium threading system classes deductions

To obtain pseudoorthogonal polynomials will expand bases p_{1}(z),...,p_{k}(z) on r control bases p_{k+1}(z),...,p_{k+r}(z), and suppose orthogonal polynomialsas

The expression (1) defines values pseudoorthogonal polynomials that have violated the orthogonality test grounds.

According to Chinese theorem rests

the polynomial can be represented in the form

Then each summand of the expression (2) represents

whereorthogonal basis nonredundant system grounds PSCW.

Substituting equation (1) in the equality (3) and taking into account that during operation no output beyond the P_{slave}(z), we obtain

Therefore, justly

Thus, on the basis of expressions (5) and using the values pseudoorthogonal polynomials, defined by the equality (1), one can calculate the values of the residues on the test groundsaccording to

Then, on the basis of the received C is achene
and values α_{k+1}(z),...,α_{k+r}(z)at the input device, error correction, it is possible to determine the difference according to the expression

If the difference is equal to zero, ie,

θ_{k+1}(z)=0,...,θ_{k+r}(z)=0,

the original polynomial A(z)∈P_{slave}(z) is permitted and does not contain errors. Otherwise, modular combination is prohibited. Then depending on the value defined by expression (7), corrects errors, ie,

where (0,...,Δα_{i}(z),...,0) - vector of errors modular code; Δα_{i}(z) - depth error for the i-th module;

Let set extended Galois field GF(2^{4}), which outlines the following reasons:

p_{1}(z)=z+1; p_{2}(z)=z^{2}+z+1, 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 R_{1}(z), R_{2}(z), p_{3}(z) - operating bases, p_{4}(z), p_{5}(z) - control base PSCW*.*

Then, according to expressionhave

P_{slave}(z)=z^{7}+z^{6}+z^{5}+z^{2}+z+1.

Full system grounds PSCW GF(2^{4}defined orthogonal bases:

B_{1}(z)=z^{14}+z^{13}+z^{2}
+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.

Using the comparability of orthogonal bases and nonredundant full system operating range, we determine the values of orthogonal bases nonredundant system bases p_{1}(z)=z+1; R_{2}(z)=z^{2}+z+1;*p*_{3}*(z)=z*^{4}*+z*^{3}*+z*^{2}*+z+1. Then*

Based on these values define all pseudoorthogonal polynomials for PSCW GF(2^{4})given the impossibility of going beyond the operating range R_{slave}(z)=z^{7}+z^{6}+z^{5}+z^{2}+z+1. The values obtained are given in table 1.

Table 1 | |

Pseudoorthogonal polynomials PSCW GF(2^{4}) | |

The basis PSCW | Pseudoorthogonal polynomial |

(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^{3}+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+1, z^{3}+z^{2}+z, z^{3}+z^{2}+1) | |

(0, 0, z^{2}z, z^{3}) | |

(0, 0, z^{2}+1, 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+1) | |

(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^{3}+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_{slave}then true

(α_{1}(z), α_{2}(z), α_{3}(z))=(α_{1}(z), α_{2}(z), α_{3}(z), B1;
^{*} _{4}(z), α^{*} _{5}(z))=(α_{1}(z), α_{2}(z), α_{3}(z), α_{4}(z), α_{5}(z)),

where the left part of the equality represented 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 expressions (6)and the right modular combination of basis p_{1}(z), p_{2}(z), p_{3}(z), p_{4}(z), p_{5}(z) extended Galois fields GF(2^{4})received at the input device for the correction of errors in polynomial system classes deductions using pseudoorthogonal polynomials. Then have a normalized value

Otherwise

The result shows that the received input of the modular combination of A(z) contains an error that must be subjected to correction. Table 2 presents the values of the error vector (0,...,Δα_{i}(z),...,0) - modular code for different values of the error syndrome for PSCW GF(2^{4}).

Table 2 | ||

The values of the error vector modular code field GF(2^{4}) | ||

The basis PSCW | The value of the vector of error | |

;
_{4}(z) | θ_{5}(z) | |

0 | 0 | (0, 0, 0, 0, 0) |

z^{3}+z+1 | Z | (1, 0, 0, 0, 0) |

z^{2}+z+1 | z^{3}+1 | (0, 1, 0, 0, 0) |

z^{3}+z | z^{2}+z+1 | (0, z, 0, 0, 0) |

z^{3}+z^{2}+1 | z^{3}+z | (0, 0, 1, 0, 0) |

z+1 | z^{2}+z+1 | (0, 0, z, 0, 0) |

z | z^{3} | (0, 0, z^{2}, 0, 0) |

z^{2} | z+1 | (0, 0, z^{3}, 0, 0) |

1 | 0 | (0, 0, 0, 1, 0) |

z | 0 | (0, 0, 0, z, 0) |

z^{2} | 0 | (0, 0, 0, z^{2}, 0) |

z^{3} | 0 | (0, 0, 0, z^{3}, 0) |

0 | 1 | (0, 0, 0, 0, 1) |

0 | Z | (0, 0, 0, 0, z) |

0 | z^{2} | (0, 0, 0, 0, z^{2}) |

0 | z^{3} | (0, 0, 0, 0, z^{3}) |

In the extended Galois field GF(2^{4}you can use the generic article number system with bases

Then

From the equation (9) shows that if A (z) ∈ P_{slave}(z), that is, ord A(z)<ord R_{slave}(z), then the older coefficients OPS, appropriate control grounds, must be equal to zero, that is,

and_{4}(z)=0 and a_{3}(z)=0

Otherwise, A(z) contains an error, therefore, for the detection and correction of errors must be calculated and_{4}(z) and a_{5}(z). Then the calculation of the senior coefficients OPS through the normalized trace is performed according to the expression

j=k+1,...,k+r;the constant recalculation of PCCW in OPS.

For the field GF(2^{4}with two reference bases p_{4}(z)=z^{4}+z^{3}+1 and R_{5}(z)=z^{4}+z+1 have

The constant values of conversion are shown in tables 3 and 4.

Table 3 | |

The constant values of the conversion module p_{4}(z) | |

θ_{4}(z) | |

1 | z^{2}+z |

Z | z^{3}+z^{2} |

z^{2} | 1 |

z^{3} | z |

T the blitz 4 | ||

The constant values of the conversion module p_{5}(z). | ||

θ | ||

θ_{4} | 1 | z^{3}+z^{2}+z |

z | z^{3}+z^{2}+z+1 | |

z^{2} | z^{3}+z^{2}+1 | |

z^{3} | z^{3}+1 | |

θ_{5} | 1 | z |

z | z^{2} | |

z^{2} | z^{3} | |

z^{3} | z+1 |

Imagine orthogonal bases in the form of coefficients OPS

If the orthogonal bases In_{i}(z) to get their views in the OPS, the result of the execution of the expression (10) determines the coefficients OPS

wherethe coefficients OPS i-th orthogonal basis taking into account the overflow (i-1)-th base. This multiplication deductions α_{i}(z) to the corresponding coefficientsis the unit bitwise and, taking into account the excess of the module P_{i}(z) as the transfer senior to the ratio of OPS α
_{i+1}(z).

Given a polynomial A(z)=z^{6}+z^{5}+z^{4}+z+1, which belongs to P_{floor}(z)=z^{l5}+1. Then in the code PSCW A(z)=(1, z+1, z^{3}+z^{2}+z+1, z^{3}+z^{2}+z, z^{3}+z). We will use the expression (10) and valuesfrom tables 3, 4 for determining the coefficients OPS control bases p_{4}(z) and R_{5}(z).

Thus, for a polynomial A(z)=(1, z+1, z^{3}+z^{2}+z+1, z^{3}+z^{2}+z, z^{3}+z) values of the older coefficients OPS is equal to:

a_{4}(z)=0, a_{5}(z)=0,

indicating the absence of the error.

Suppose that received combination error occurred on the first base and its depth is equal to Δa_{1}(z)=1. Then we have:

A*(z)=(0, z+1, z^{3}+z^{2}+z+1, z^{3}+z^{2}+z, z^{3}+z).

According to expression (1) and the data presented in table 1, we obtain the following pseudoorthogonal polynomials A_{i}(z):

A_{1}(z)=(0, z+1, 0, z^{3}+z^{2}+1, z^{3}+z^{2}+z);

A_{2}(z)=(0, 0, z^{3}+z^{2}+z+1, z^{3}, z^{2}+z).

Using the expression (6) determine the values of the residuals on control basis:

According to expression (7) is defined:

θ_{4}(z)=(z^{3}+z^{2}+z-z^{2}+1)mod(z^{4}+z^{3}+1)=z^{3}+z+1;

θ_{5}(z)=(z^{3}+z-z^{3})mod(z^{4}+z+1)=z.

Thus

±
_{4}(z)=z+z^{3}+z^{2}+z^{2}+z=z^{3}

α_{5}(z)=z^{3}+1+z^{3}+z^{2}+z+1+z+z^{3}+z^{2}+z+z^{2}=z^{3}+z^{2}+z

Table 5 | ||

The dependence of the values of the coefficients OPS location and depth errors for the field GF(2^{4}) | ||

The magnitude of the error | The coefficients OPS | |

a_{4}(z) | a_{5}(z) | |

Δα_{1}=1 | z^{3} | z^{3}+z^{2}+z |

Δα_{2}=1Δα _{2}=z | z^{3}+z+1 | z^{3}+z^{2} |

z^{3}+z^{2}+z | z^{3}+z | |

Δα_{3}=1Δα _{3}=zΔα _{3}=z^{2}< / br>Δα _{3}=z^{3} | z^{2}+1 | z^{3}+z^{2}+z |

z^{3}+z | z^{3}+z^{2}+z+1 | |

z^{3}+z^{2} | z^{3}+z^{2} | |

1 | z^{3}+z | |

Δα_{4}=1Δα _{4}=zΔα _{4}=z^{2}< / br>Δα _{4}=z^{3} | z^{2}+z | z^{3}+z^{2}+z |

z^{3}+z^{2} | z^{3}+z^{2}+z+1 | |

1 | z^{3}+z^{2} | |

Z | z^{3}+z+1 | |

Δα_{5}=1Δα _{5}=zΔα _{5}=z^{2}< / br>Δα _{5}=z^{3} | 0 | z |

0 | z^{2} | |

0 | z^{3} | |

0 | z+1 |

From table 5 in accordance with the α_{4}(z)=z^{3}and α_{5}(z)=z^{3}+z^{2}+z is selected the error rate, which is formed with A*(z)=(0, z+1, z^{3}+z^{2}+z+1, z^{3}+z^{2}+z, z^{3}+z), which equals A(z)=(1, z+1, z^{3}+z^{2}+z+1, z^{3}+z^{2}+z, z^{3}+z).

Error in modular code fixed.

The technical result of the invention is the provision of improved performance and correction of errors.

Increased performance is achieved in that the device comprises a positional accumulating adder whose output is the output of the device, and the synchronization unit containing the memory block is constant, the group of items, shift register, a group of inputs which is the input device, and a control input connected to the first output of the synchronization unit, the first inputs of elements And groups merged and connected to you is an ode LSB shift register, and second inputs connected to the corresponding data outputs of the memory block constants, the control input of which is connected to the second output of the synchronization unit, the outputs of the elements And groups connected to the corresponding inputs positional accumulating adder.

Figure 1 shows the structure of the device for converting the number of PCCW in the positional correction code errors.

The device comprises a shift register 1, the synchronization unit 2, unit 3 memory constants, a group of items And 4, the positional accumulating adder 5, input 6 device block 7 error is detected, block 8 storage of data, corrective modulo two 9, the output 10 of the device.

The number in the code PSCW through the inlet 6 is recorded in the shift register 1, which is a set of sub-registers for storing deductions for each of the grounds PSCW. In the future, the case 1 is considered as a single register, the contents of which in each step moves to the right towards the "younger" digits by one digit.

Unit 3 memory constants containsconstant bit

The operation of the device occurs in cycles. Each bar is issuing another constant of the memory block constants 3 value 2^{i-1}·B_{ji}in the case of open items And group the s -
adding it to the content positional accumulating adder 5 working modulo p, and shift the source code of the number in register 1 by one digit to the right. The operation ends after m clock cycles and does not depend on the value of the original number. The time of operation of the known device depends on the number and ranges from 0 to R.

Concurrently with entering the number in the code PSCW on the shift register, it goes to the block error detection 7, where the error is detected on the trail of a polynomial. If the number is correct, its trace is normalized and equal to zero. Otherwise, it is not equal to zero and the input of corrective modulo two 9 comes Δ_{cor}previously listed in block 8 data storage. In a correction modulo two 9 is the sum modulo two of the values A* and Δ_{cor}therefore, on the output device 10 receives fixed combination.

Thus, the introduction of the device unit 7 error is detected, block 8 data storage and corrective modulo two 9 allows the classical way, and with the help of minimal polynomials not only detect, but also to correct the error.

Unit 7 error detection is a three-layer neural network (figure 2). The input layer consists of 15 neurons, distributed according to the dimension of the bit grid modules-1-2-4-4-4.
These neurons carry out the bifurcation of the input vector ((α_{1}(z), α_{2}(z), α_{3}(z), α_{4}(z), α_{5}(z)), represented in binary form. Moreover, neuron 11 is designed for the distribution of zero dischargefirst base. Neurons 12, 13 are designed to distribute zeroand the firstbits of the second base, respectively. Neurons 14-17 are used for reception and distribution,,andbits of the third base respectively. For redistribution zerothe firstthe secondand the thirdbits of the fourth Foundation of the p_{4}(z) are the neurons 18, 19, 20, 21, respectively. Neurons 22, 23, 24, 25 are used to receive zerothe firstthe secondand the thirdbits of the fifth Foundation of the p_{5}(z), respectively.

The second layer is designed to calculate the normalized footprint and consists of eight neurons, distributed in soo is Ted dimensionality bit nets coefficients (α
_{i}(z), i=4,5) OPS 4-4. Neurons 26-33 of the second layer performs a basic operation of summing modulo two values of the bits of the outputs of the respective neurons of the first layer, and the neuron 26 of the second layer receives signals from the neurons 11, 12, 14, 15, 18 input layer to neuron 27 of the second layer with neurons 11, 12, 13, 15, 16, 19 input layer to neuron 28 of the second layer with neurons 12, 14, 17, 20 input layer to neuron 29 of the second layer with neurons 11, 13, 14, 21 input the layer. These four neuron of the second layer are used to calculate the constants conversion θ_{4}. The neuron 30 second layer receives signals from the neurons 12, 13, 15, 17, 22 input layer to neuron 31 of the second layer with neurons 11, 13, 14, 15, 17, 23 input layer to neuron 32 of the second layer with neurons 13, 15, 24 input layer to neuron 33 of the second layer with neurons 12, 14, 16, 25 of the input layer. These four neuron of the second layer are used to calculate the constants conversion θ_{5}.

The third layer is used to evaluate senior coefficients OPS and also consists of eight neurons, distributed in accordance with the dimension of the bit nets coefficients (α_{i}(z), i=4,5) OPS 4-4. Neurons 34-41 third layer performs the basic operation of summing modulo two values of the bits of the outputs of the respective neurons of the second layer, and the neuron 34 of the third with the HHS receives signals from the neuron 28 of the second layer,
the neuron 35 of the third layer with neurons 26, 29 of the second layer, neuron 36 of the third layer with neurons 26, 27 of the second layer, neuron 37 of the third layer with neuron 27 of the second layer. These four neuron of the third layer are used to calculate OPS and_{4}. The neuron 38 of the third layer receives signals from the neurons 27, 33 of the second layer, neuron 39 of the third layer with neurons 26, 27, 30, 33 of the second layer, neuron 40 of the third layer with neurons 26, 27, 31 of the second layer, neuron 41 of the third layer with neurons 26, 27, 32 of the second layer. These four neuron of the third layer are used to calculate OPS and_{5}.

The output signal of the third layer are received at the inputs of the block 8 data storage.

1. A device for converting the number of polynomial system classes deductions in the positional correction code errors containing positional accumulating adder, the synchronization block, the memory block is constant, the group of items, shift register, a group of inputs which is the input device, and a control input connected to the first output of the synchronization unit, the first inputs of elements And groups merged and connected to the output LSB shift register, and the second inputs connected to the corresponding data outputs of the memory block constants, the control input of which is connected to the second output unit Singh is oneseli, the outputs of the elements And groups connected to the corresponding inputs positional accumulating adder, wherein the inputs of block error is detected, the input of which is connected to the input device, the output unit error is detected connected to the inputs of the block data storage, which is designed to store the dependencies of the values of the coefficients of the generalized article system from the location and depth errors for a given Galois fields, inputs corrective modulo two is connected to the output of the block data storage and outputs positional accumulating adder, the output adjustment of the modulo two is the output device.

2. The device according to claim 1, characterized in that the block error detection is a three-layer neural network, the input layer which contains fifteen neurons, the second layer with eight neurons in the third layer with eight neurons and the inputs of the first neuron of the second layer are connected to the outputs of the first, second, fourth, fifth, eighth neurons in the input layer, and its outputs connected to the inputs of the second, third, sixth, seventh, eighth neurons of the third layer; the inputs of the second neuron of the second layer are connected to the outputs of the first, second, third, fifth, sixth, ninth neurons in the input layer, and its outputs connected to input what am third, fourth, fifth, sixth, seventh, eighth neurons of the third layer; the inputs of the third neuron of the second layer are connected to the outputs of the second, fourth, seventh, tenth of neurons in the input layer, and its outputs connected to the inputs of the first, seventh, eighth neurons of the third layer; a fourth inputs of the neuron of the second layer are connected to the outputs of the first, third, fourth, eleventh neurons in the input layer, and its outputs connected to the inputs of the second, fifth, sixth, eighth neurons of the third layer; the inputs of the fifth neuron of the second layer are connected to the outputs of the second, third, fifth, seventh, twelfth neurons in the input layer, and its outputs connected to the inputs of the sixth neuron of the third layer; the inputs of the sixth neuron of the second layer are connected to the outputs of the first, third, fourth, fifth, seventh, thirteenth neurons in the input layer, and its outputs connected to the inputs of the seventh neuron of the third layer; the inputs of the seventh neuron of the second layer are connected to the outputs of the third, fifth, fourteenth neurons in the input layer, and its outputs connected to the inputs of the eighth neuron of the third layer; the inputs of the eighth neuron of the second layer are connected to the outputs of the second, fourth, sixth, sixteenth neurons in the input layer, and its outputs connected to the inputs of the fifth, sixth neurons of the third layer.

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SUBSTANCE: device uses neural-network technologies and polynomial residuals system, wherein as system base minimal polynomials p_{i}(z), where input=1,2,...,n, are utilized, determined in expanded Galois fields GF(2^{V}), while device has clock counter, two blocks for calculating sums of paired results of multiplication by arbitrary base, error correction block, modular adder and block for calculating sums of paired results of multiplication based on control base.

EFFECT: decreased hardware requirements, improved speed of operations.

2 dwg, 3 tbl

FIELD: computer science.

SUBSTANCE: device has harmonic signal generator, controlled phase changers, means for measuring phase of harmonic signal, phase changers for fixed phase values, transformers of binary number code to unary in accordance to first and second sub-modules, coder and table calculation means.

EFFECT: lower costs.

3 dwg

FIELD: computer engineering, possible use in modular neuro-computer systems.

SUBSTANCE: in accordance to invention, neuron network contains input layer, neuron nets of finite ring for determining errors syndrome, memory block for storing constants, neuron nets for computing correct result and OR element for determining whether an error is present.

EFFECT: increased error correction speed, decreased amount of equipment, expanded functional capabilities.

1 dwg, 3 tbl