NOISE IMMUNITY OF THE ALGEBRAIC GEOMETRIC CODES

: Linear block noise-immune codes constructed according to algebraic curves (algebraic geometric codes) are considered, their design properties are evaluated, algorithms of construction and decoding are studied. The energy efficiency of the transmission of discrete messages by M -ary orthogonal signals in the application of algebraic geometric codes is studied; the achievable energy gain from the use of noise-immune coding is estimated. The article shows that in discrete channels without memory it is possible to obtain a significant energy gain, which increases with the transition to long algebraic geometric codes constructed from curves with a large number of points relative to the genus of the curve. It is found that the computational complexity of implementing algebraic geometric codes is comparable to other known noise-immune codes, for example, Reed-Solomon codes and others. Thus, high energy efficiency in combination with the acceptable computational complexity of implementation confirms the prospects of algebraic geometric codes use in modern telecommunication systems and networks to improve the noise immunity of data transmission channels.

The aim of this paper is to study some algorithms for constructing and decoding algebraic geometric codes and to estimate the energy efficiency achieved. In addition, the paper presents the results of comparative studies of the complexity of the practical implementation of noise-immune coding and decoding using algebraic geometric codes and other (the most well-known and common) codes. The results of this research can be used to improve various code-based methods of information security [16][17][18][19][20][21][22][23][24][25][26].
The paper is structured as follows. In Section 2, the basic concepts and definitions of algebraic geometric codes are given, the constructive characteristics of the codes are evaluated, and their asymptotic properties are studied. Section 3 is devoted to the development and study of computationally efficient procedures for encoding and decoding of algebraic geometric codes. In particular, we have presented several options for constructing codes (in a systematic and nonsystematic form) and a simple decoding algorithm. Section 4 examines the energy efficiency of algebraic geometric coding in the transmission of discrete messages by M-orthogonal signals. In Section 5, we conduct a comparative analysis of the complexity of the implementation of the encoding computing@computingonline.net www.computingonline.net and decoding algorithms. The conclusions of the work summarize the obtained results and show the promising areas for further research.
The number N of the points of the curve X over ( ) GF q is bounded above by the Hasse-Weil expression [1][2][3]: 2 1 N q g q     . Table 2 gives the upper bound for the number of points of the curve over a finite field. The limit values of the number of points of smooth curves are summarized in Table 3.
The Singleton boundary indicates the maximum achievable code distance for the given code parameters ( , , ) n k d and is written in the form For the code ( , , ) n k d , consider the following parameters: is the relative speed of the code, as the fraction of information symbols in the transmitted data; In [3], the asymptotic boundary of algebraic geometric codes is given by Fig. 1 shows the asymptotic boundaries: 1 is the Singleton boundary; 2 is the Varshamov-Gilbert boundary; 3 is the boundary of the algebraic geometric codes.
The above dependences show that as the power q of the alphabet of code symbols increases, the asymptotic properties of algebraic geometric codes improve. Obviously, for large q , these codes lie above the Varshamov-Gilbert boundary, which indicates high potential characteristics.

Figure 1 -Asymptotic properties of algebraic geometric codes
The constructive code characteristics of algebraic geometric codes over curves of genus 0 g  , 1 g  , 3 g  , 6 g  , over (4) GF are summarized in Table 4. The corresponding constructive estimates of the code parameters for curves of the various genus values

CODING AND DECODING BY ALGEBRAIC GEOMETRIC CODES
We consider the coding operations by algebraic geometric codes for the general case. In other words, for curves defined in the projective space (1) be N of the joint solutions of equations (1) for the points of the curve X .
We fix the divisor D of the curve X and the set of rational functions associated with the divisor D , i.e., the set consisting of zero and functions 0 where 0 1 , ,..., w F F F are forms with the same degree In other words, Let  be the degree of the divisor class, 1 g    , then the mapping : of the algebraic geometric code with the constructive characteristics (here and below the symbol ... is used to denote the matrix).
To form the code word   Obviously, the formation of the code word is carried out by an iterative procedure, allowing at each step of the algorithm to generate the appropriate code symbol.

CODING IN A SYSTEMATIC MANNER THROUGH A CHECK MATRIX
of the algebraic geometric code with the constructive characteristics The algebraic geometric code from the curve X over ( ) GF q constructed through the verification matrix H is a linear code consisting of all words To form the code words of the algebraic geometric code given in this way on the space curves, we use the matrix inversion techniques [4][5][6].
We divide the code word   In the matrix representation, the last notation is equivalent to the expression To find the values of r n k   verification symbols, using matrix inversion methods, we write where      Since the placement of verification items is usually known and fixed, the inverse matrix for the system of equations (5)

DECODING OF ALGEBRAIC GEOMETRIC CODES
The equality holds: Suppose that when transmitting via a channel with errors the codeword is distorted, then the error vector will be denoted by   Taking the latter into account, formula (11) can be rewritten as: We now return to the consideration of polynomial (9). We multiply it by an arbitrary monomial 0 and carry out analogous arguments. By analogy with (10)  ... ... ...
If the number of unknowns z in the polynomial of error locators is less than the number of elements of the syndrome sequence, the system of linear equations (12) is solvable. The complexity of its solution, for example, by the Gaussian method is equal to The solutions of system (12) give the values of the unknown coefficients of the polynomial (9) of the error locators

ENERGY EFFICIENCY OF ALGEBRAIC GEOMETRIC CODING
To estimate the energy efficiency of the algebraic geometric coding, consider the option of where  is the signal-to-noise ratio for the M -th symbol, 2 m M  , 2  is the normalized signal-tonoise ratio per binary unit, 2 m    . Fig. 3 shows the dependence of the probability of erroneous reception of the M -th symbol in the coherent reception of orthogonal signals.
The transmission of M orthogonal signals makes it possible to obtain a significant gain of noise immunity for a fixed signal-to-noise ratio, or a significant energy gain with a fixed error probability (for each symbol). As the power of the ensemble of signals increases, this gain goes up, too.
If the decoder corrects errors, then the probability of erroneous decoding of the block is If we accept the assumption of a random occurrence of 2 1 t  and more errors as a result of erroneous decoding of the code word, then the mathematical expectation of the number of erroneous information symbols at the decoder output is determined by the expression [4-6] (16) and the probability of erroneous decoding of the information symbol is The use of codes that detect and correct errors leads to an increase in the redundancy of the transmitted data. If we fix the energy of the message transmitted to the channel, then the energy per one symbol will decrease proportionally to the redundancy introduced.
To calculate the dependencies of the error probability on the symbol at the output of the decoder (14)(15)(16)(17), taking into account the redundancy introduced, the signal-tonoise ratio  in expression (13) is reduced by R k n  times. Let's consider a variant of transferring discrete messages by 4 orthogonal signals. The transmitted messages are encoded by an algebraic geometric code constructed above the field GF(4) (the selected code parameters are highlighted in color in Table 4). Figure 4 shows the dependencies of the error probability of the 4-th symbol on the normalized signal-to-noise ratio when four orthogonal signals are coherently received using noise-immune algebrageometric codes. A relationship marked as "M=4" corresponds to an uncoded transmission. The dependence marked as (5,3,3) corresponds to an algebraic geometric code along a curve of genus 0 g  whose code characteristics lie on the Singleton boundary. This code with the maximum distance separable codes (MDS code) is the extended Reed-Solomon code. The greatest energy gain in algebraic geometric codes (including MDS codes) is given at the speed 2 / 3 R  . The dependencies presented in Figure 4 show the advantages of using algebraic geometric codes for noise-immune message transmission. Thus, with the error probability per symbol 5 10 c P   , the application of the code (9,6,3) gives an energy gain of 0,6dB compared to the uncoded message transmission and 0,2dB in comparison with the MDS code.
Let's consider a variant of transferring discrete messages by 8 orthogonal signals. The transmitted messages are encoded by an algebraic geometric code constructed over the field GF (8). The constructive code characteristics of the selected codes are highlighted in color in Table 5. Figure 5 shows the dependence of the error probability of the 8th symbol on the normalized signal-to-noise ratio when coherently receiving 8 orthogonal signals using noise-immune algebra-geometric codes. The relationship marked as "M=8" corresponds to the uncoded transmission. With the error probability per symbol is 6 10 c P   , the use of the code (23,14,7) gives an energy gain of 2dB in comparison with the non-coded message transmission and 0,8dB in comparison with the MDS code. Let's consider a variant of transferring discrete messages by 16 orthogonal signals. The transmitted messages are encoded by an algebraic geometric code constructed over the field GF (16). The constructive code characteristics of the codes selected for the evaluation are outlined in Table 6. Figure 6 shows the dependencies of the error probability of the 16th character on the normalized signal-to-noise ratio when the coherent reception of 16 orthogonal signals using noise-immune algebrageometric codes. The relationship marked as "M=16" corresponds to the uncoded transmission. With the error probability per symbol  Let's consider the transfer variant of discrete messages by 32 orthogonal signals. The transmitted messages are encoded by an algebraic geometric code constructed over the field GF(32). The selected code characteristics are outlined in Table 7. Figure 7 shows the dependence of the error probability of the 32nd symbol on the normalized signal-to-noise ratio when the coherent reception of 32 orthogonal signals using noise-immune algebraic geometric codes. The relationship marked "M=32" corresponds to the uncoded transmission. With the error probability per symbol 9 10 c P   , the use of the code (99,65,29) gives an energy gain of 4,5dB in comparison with the non-coded message transmission and 1dB in comparison with the MDS code.
Let's consider the transfer variant of discrete messages by 64 orthogonal signals. The transmitted messages are encoded by an algebraic geometric code constructed over a field GF(64) with the parameters highlighted in Table 8. Figure 8 shows the dependencies of the error probability of the 64th symbol on the normalized signal-to-noise ratio when coherently receiving 64 orthogonal signals using noise-immune algebraic geometry codes. A relationship marked as "M=64" corresponds to an uncoded transmission. With the error probability per symbol, the application of the code (164,110,49) gives an energy gain of 6dB as compared to noncoded message transmission and 0,8dB in comparison with the MDS code.
As follows from the dependencies presented in Figures 4-8, the use of algebraic geometric codes for increasing the noise immunity of message transmission in discrete channels without memory leads to a significant energy gain. Their use also makes it possible to significantly reduce the probability of error per symbol with a fixed signalto-noise ratio per one transmitted bit. The energy gain increases with the transition to codes constructed from curves with a large number of points relative to the genus of the curve.

COMPARATIVE EVALUATION OF THE IMPLEMENTATION COMPLEXITY OF CODING AND DECODING
Let's make a comparative evaluation of the complexity of the implementation of the encodingdecoding procedures for the considered codes in comparison with the known schemes. The complexity of the encoding-decoding procedures (as well as the estimation of the energy gain of algebraic geometric coding) is evaluated in comparison with the Reed-Solomon codes.
If the code is given by the generator matrix G, the systematic encoding procedure is equivalent to multiplying the information word I by this matrix, that is, c=IG, where c is the code word. In this sense, the complexity of algorithms for systematic coding of algebraic geometric codes and Reed-Solomon codes is practically equivalent.
Reed-Solomon codes are a subclass of BCH codes, therefore, the same methods as for BCH codes are applied to their decoding. One of the most effective algorithms for algebraic decoding of BCH codes is the Berlekamp-Messi algorithm and its modifications (improvements). It is known [6] that the Berlekamp-Messi algorithm contains the number of multiplications, of order t 2 , or, formally, the complexity of the algorithm O(t 2 ). For large t, we use the accelerated Berlekamp-Massey algorithm, which makes it possible to reduce the computational complexity of the algorithm. Even more effective, in terms of computational complexity, is the recurrent algorithm of Berlekamp-Messi. The asymptotic complexity of decoding the Reed-Solomon codes, in this case, does not exceed O(nlog 2 n), and is very close to the value of O(nlogn).
Algorithms for decoding algebraic geometric codes were developed in [7][8][9]. So, in [7], a decoding algorithm is proposed, the complexity of which is determined by the value O(n 3 ). Further development of the decoding procedure in [8] made it possible to reduce the computational complexity (shown by the example of Hermite curves) to O(n 7/3 ). In [9], the algorithm of decoding, of complexity O(n 2 ), which allows parallelization of computations (on n processors) is considered. Obviously, the existing algorithms for decoding algebraic geometric codes are comparable in computational complexity with the algorithms for decoding BCH codes.

CONCLUSION
A lot of studies have shown that algebraic geometric codes boast very constructive characteristics. In particular, the dependences given in Fig. 1 indicate that as the power of the alphabet increases, the code characteristics improve. With a large length, algebraic geometric codes lie above the Varshamov-Gilbert boundaries, which indicates high potential characteristics. We obtained code characteristics for various curves over finite fields GF(2 m ), m=2,…,6 (Tables 4-8).
As the studies show, the practical implementation of the algorithms for encoding and decoding algebraic geometric codes is reduced to simple and computationally efficient operations on finite fields. We presented several options for constructing codes (in a systematic and non-systematic form), and the simple decoding algorithms. The implementation of these algorithms does not require significant computational costs: as shown by the performed analysis, the complexity of encoding and decoding is comparable to other known classes of codes.
To evaluate the energy efficiency of algebraic geometric coding, we considered the option of discrete messages by M-orthogonal signals transmitting. As follows from the dependences shown in Figures 4-8, the use of algebraic geometric codes in discrete channels without memory leads to a significant energy gain. The energy gain increases with the transition to long codes constructed from curves with a large number of points relative to the genus of the curve.
The high energy efficiency of algebraic geometric coding in combination with an acceptable complexity of practical implementation allows us to talk about the possibility of constructing effective noise-immune systems based on the use of such codes [13][14][15]. Development and implementation of practical recommendations on the direct use of algebraic geometric codes in modern telecommunication systems and networks is a promising direction for further work. In addition, a prospective direction of further research is the argumentation of practical recommendations concerning the implementation of the introduced method and the ways of its use in different mechanisms of information security of telecommunications networks and systems.