Selection and justification of polynomial generators for encoding telemetric information in Autonomous well systems

. In this paper, the issue of effective implementation of wireless packet data transmission for oil production systems is raised in order to form a competence-based approach to solving key problems of field development in conditions of disequilibrium and heterogeneity of geological and field data. Using the proof of the theorem on the definition of the convolutional code over the Galois field, boundary conditions are obtained, the creation of which is necessary in order to implement high noise immunity when transmitting telemetric signals from wells to the “day surface”. The algorithm presented in the study is based on the application of equations of various orders to level out the significant heterogeneity of field data, which, in conditions of a widespread increase in the number of wells, has a significant impact on making effective and high-quality management decisions. The results of the study can also serve to determine the correction capabilities provided by digital signal processing computing systems in autonomous downhole telemetry modules.


Introduction
One of the basic control tasks in oil production is to ensure noise immunity of wireless communication channel and reliability of information transmitted on productive strata [1].When oil and gas wells are explored using submersible telemetry systems, one of the most crucial challenges is to ensure reliable transmission of well information to the so-called 'daylight surface'.Transmitting information via an electromagnetic communication channel is a particular challenge.Telemetry information represents data on temperature, pressure, flow rate, water content in a mixture, etc.
To ensure interference-free reception of information, high-rate transmitters, ultrasensitive receivers are used.They reduce the range to low and ultra-low frequencies and duplicate the transfer of broadband signals.In order to detect, localize and correct computational errors likely to spring during digital signal processing, various methods of error-correcting coding are used to offer such assistance [2].Anti-jamming codes fall into codes that correct single-bit or burst errors.Single-bit errors are errors, each pair of which is separated by at least n symbols.In practice, the properties of communication channels are such that errors are clustered, resulting in burst errors.The authors mostly deal with methods to reduce impacts of burst errors.Hence, once received, the likely burst errors were transformed into groups of independent random errors, thereby increasing the efficiency of noise-resistant coding.The paper is concerned with the ways tailor-made polynomials are created for convolutional codes over Galois fields.The findings are used to effectively estimate code distances of a shortened code and, on this basis, to effectively control the development of oil reserves from productive strata [3], which underpins the most urgent issue of increasing the efficiency of producing hard-to-recover deposits [4-9].

Methods and materials
To ensure the performance indicators of submersible borehole telemetry system, the authors in investigate the issues of selection and justification of electromagnetic channel devices, primary and secondary multiplexing, autonomous power supply systems, etc.
Herewith, the authors pay special attention to polynomial generators created to represent the convolutional code ( 0 ,  0 ), estimate the value of the distance in this code with a rate of  = 1/.Convolutional codes are continuous sequences of single elements, not divided into blocks.In such codes, redundant bits are placed in a certain order between information bits.The distance is estimated using the code distance theorem for a shortened cyclic code: if a polynomial () =    3 +  −1  −1 + ⋯ +  1  +  0 generates a cyclic code (, ) over the field GF(2  ) with code distance , the resulting shortened cyclic code ( с ,  с ) with polynomial generator   () and code distance  с will be  с ≥ .
In this case, the minimum code distance   is a minimum number of symbols in which two codewords differ from the entire set of codewords of the convolutional code over the constraint length: Expression (1) determines the number of symbols.The free code distance   of the convolutional code is the minimum number of symbols in which two codewords differ from the whole set of codewords of the convolutional code over a length of   =  + ∑    =1 () symbols.It is known from that choosing a convolutional code with threshold decoding, the minimum code distance is critical.For other decoding methods, the free code distance is decisive.Let us introduce the concept of an optimal convolutional code.These are convolutional codes found by exhaustive methods that have the maximum value of the minimum or free code distance.Note that this happens with the minimum value of a code constraint.Assume that the source cyclic code (, ) over the field GF (2  ) is used to construct a convolutional code ( 0 ,  0 ) over the field GF (2)and has a code distance .
Then the shortened cyclic code obtained from the source cyclic code by imposing constraints on the set of information polynomials will have the code distance  с ≥ .This statement is consistent with the theorem on the definition of convolutional code over Galois field: If the polynomial () =    3 +  −1  −1 + ⋯ +  1  +  0 generates a cyclic code (, ) over the field GF (2  ), the resulting shortened cyclic code ( с ,  с ) will completely define the convolutional code ( 0 ,  0 ) over the field GF (2) with a rate  = 1/ and the polynomial generator   () of the form () =    3 +  −1  −1 + ⋯ +  1  +  0 , where () =   ().
Let  1 ,  2 , … ,   be the basis of elements of the field GF (2  ).Then given that: An element of the field GF (2  ), where   is an element of the field GF (2), is mapped to an -digit vector  = ( 1 ,  2 , … ,   ).This mapping converts linear to non-linear codes.Recall that if redundant symbols of the code sequence are generated by applying some linear operations to their information symbols, the code is called linear.Otherwise, the code is non-linear.
As the basis,  linearly independent vectors are chosen.So the field GF (2  ) can be displayed in the field GF (2) using various bases in  2  −1  variants.It should be borne in mind that the determinant composed of the basis vectors must be nonzero.Otherwise, the mapping is not realizable.For example, for an element of the field GF ( 22 ), the basis consists of  1 and  2 alone with  2 2 −1 2 = 3 choices: Where   are elements of the field GF (2 ^ 2),  = 0,1,2.Determinants composed of elements of a basis of the form (2) are nonzero, so the field GF( 22 ) receives the following mapping: Replacing the basis affects the weight spectrum and the minimum code weight.In this case, the minimum weight can only be changed upwards.The weight spectrum of the cyclic code  can be described using the following relationship: Where   is the number of codewords of weight  in the code .

Results and Discussion
Consider the above for a cyclic (7,2)-code over the field GF( 23 ) with polynomial generator: When choosing a basis : According to expression (4), the weight spectrum of the code  will be the spectrum   = 2 13  8 + 3 11  10 + 3 9  12 +  7  14 .
If we choose as a basis We get: =  15  6 + 3 11  10 + 5 9  12 .(10) Expressions ( 7) and ( 9) demonstrate that for the first choice of the basis Λ 1 , the minimum distance of (21.6)-code over the field GF(2)  ′ = 8, and in the case of Λ 2 −  ′′ = 6.Consequently, based on the above and the theorem on the code distance of the shortened cyclic code, we can write: Where   is the code distance of the shortened cyclic code mapped to the binary code.By estimating the code distance of the binary code obtained thereby, it is possible to determine the code distance of the convolutional code ( 0 ,  0 ) over the field GF(2).Moreover, its polynomial generator is obtained based on the theorem on the definition of a convolutional code over the Galois field.
Proof of correctness.According to theorems on the coding distance of shortened cyclic code and on the definition of convolutional code over the Galois field, the degree  of the polynomial generator () for the cyclic code (, ) over the field GF(2) determines the length of the shift register in the convolutional encoder.This means that if the length of the codeword  in the cyclic code is greater than the length of the constraint   of the convolutional code, then according to the theorem on the code distance of shortened cyclic code and expressions (2), the distance of the convolutional code will be provided at least by the distance of the cyclic code, which was to be demonstrated.
An algebraic method for determining the coding distance of a cyclic code is to find the maximum-length sequences of roots of its polynomial generator.The roots represent the elements of the field over which the given code exists.A cyclic code (, ) over the field GF(2  ) can be generated by several polynomial generators ().Therefore, the problem comes up as to choosing the best among these polynomials, i.e. the one that would provide the construction of an optimal convolutional code over the field GF (2).

Conclusion
Thus, based on the theorem proved on the definition of convolutional code over Galois field, it is possible to obtain polynomial generators for cyclic codes over the field GF(2) with a rate  = 1/, using the polynomial generators for a cyclic code over the field GF(2  ).The prospect of the investigations considered herein is directly related to the possibilities of ensuring high noise immunity when transmitting telemetry signals from wells to the "daylight surface".
The research was carried out in the Scientific, Innovation and Educational Center of Digital Technologies in the Oil and Gas Industry at the project office of USPTU branch in Oktyabrsky.