DEVELOPMENT OF BINARY INFORMATION COMPRESSION METHODS BASED ON THE BINOMIAL NUMERICAL FUNCTION

The application of data compression methods is an effective means of improving the performance of information systems. At the same time, interest is aroused to the methods of compression without information loss which are distinguished by their versatility, low needs of costs during implementation


Introduction
Existing and newly developed information systems are of distributed nature and must transfer and store huge amounts of information and perform frequent and complex queries to databases [1][2][3]. Insufficient bandwidth of communication channels, low speed of information storage devices, and requirements to ensuring specified levels of fail-safety and error-free transmission are the factors that limit performance gain in such systems [3][4][5].
One of the most effective measures taken to speed up information processing and reduce the cost of computer systems and the components that form the basis of present-day control systems and telecommunication consists in the application of methods and algorithms of data compression [1][2][3].
Development of compression methods and algorithms that would have the following properties and characteristics is of particular scientific and practical interest [1][2][3]: -high compression ratio and simplicity of the compressed data model; -high performance when compressing and recovering information; -be undemanding to computing resources of terminal devices; -adaptability to the type of processed data and their characteristics; -the ability to control errors when compressing and recovering data.
In this regard, the development and implementation of compression methods and algorithms with the above properties is an urgent problem of scientific and practical value.

Literature review and problem statement
Almost any data arrays and sequences have their own structural features. It is pointed out in [6,7] that taking into account these features can significantly improve the efficiency of data compression methods based on a combinatorial approach to their representation. Although the numbering of equilibrium combinations is carried out in [6] based on binomial coefficients, binomial numbers are not considered at all in the structure of processed combinations. This leads to more cumbersome calculations when finding numbers and does not provide an extension of the functionality of the proposed methods of compressive coding. The problem of compression is studied in [7] for information sequences of limited application which narrows the scope of the proposed methods. Similar to [6], binomial numbers are not used in [7] for compression and restoration.
Implementation of the combinatorial approach to compressive coding involves the application of structural positional numbering systems that differ from the conventional binary notation by their complexity and heterogeneity. In structural numbers generated by structural positional numbering systems, there are complex relationships between the weight coefficients of numbers and their positions in the numbers themselves. On the one hand, such complex connections between weights of numbers and their positions hamper arithmetic operations on them, however, on the other hand, they expand their functionality for self-control, encryption, and data compression. When designing various information systems and solving specialized information problems, such structural numbering systems as Fibonacci, factorial, and binomial numbering systems are already widely used [8][9][10].
To solve information problems including data coding, it is proposed in [8] to use Fibonacci numbering systems. But in view of their simpler structure, the Fibonacci numbers have significantly lower noise immunity and do not ensure effective error detection during information transformations. The application of a factorial numbering system is considered in [9]. This study is limited to a range of problems with permutations that do not include information compression problems.
Binomial numbering systems, in particular those dealing with a binary alphabet [10,11], are of considerable interest and broad prospects. Being more complex in their structure, binomial numbering systems have great capabilities to detect and correct errors in information sequences, generate a wide range of combinatorial objects. Besides, algorithms of obtaining binomial numbers are quite simple and there are many classes of code combinations in the structure of which binomial numbers are found [6,7,11].
Description of binomial numbering systems and the binomial numbers generated by these systems is given in [10]. However, their application is limited only to organizing binomial counting. Issues of information compression using binomial numbering systems are not touched upon at all.
Only conceptual approaches to the application of binary binomial numbering systems in information control systems are given in [11] including those for compression of binary data. A significant compression effect is observed when using a binomial numeric function to obtain binary numbers of compressible information sequences. In this case, binary binomial numbers are used as intermediate combinations in the transition to numbers. This greatly simplifies the application of the binomial numerical function and expands possibilities of controlling the transformation correctness.
Binomial compression of combinations is considered in [12] but just based on binary binomial numbers. Such compression coding is fast but does not make it possible to obtain high compression ratios. Obviously, the studies do not fully reveal in full the potential of binomial numbering systems and the possibility of implementing higher compression based on a binomial numerical function.
An ideological approach to the application of binomial compression based on a numerical function is revealed in [13]. However, there is no theoretical justification and mathematical models of compression in this study that would represent a ground for further development of binomial algorithms, systems, and compression devices.
It should be pointed out that a system of numerical coding the binary combinations with constant weight is given in [14]. But like [4], binomial numbers are not used at all when numbering which in general does not simplify the implementation of compression or reduce the hardware, software, and time costs.
Thus, the following conclusions can be drawn based on analysis of [4][5][6][7][8][9][10][11][12]: -application of binary binomial numeration systems determines the versatility of compressive coding due to taking into account the only and widespread limitation on the number of ones in binary sequences; -determination of binomial numbers in the data structure should be distinguished as a separate stage in compression which will reduce costs of numbering sequences and control errors due to the noise immunity inherent in binomial numbers; -in order to increase the degree of binomial compression, it is necessary to use a numerical function of the binary binomial numbering system which assigns corresponding numbers to binary sequences; -for the further development of binomial compression, it is necessary to construct mathematical models and algorithms for the compression of binary data based on a binomial numerical function.

The study objective and tasks
The study objective implies the elimination of information redundancy of binary sequences with cutting time and hardware and software costs. This will improve the performance of information-and-control and telecommunication systems handling both binary combinations with a given limitation on the number of units and information sequences of general form.
To achieve the study objective, the following tasks were formulated: -construct a mathematical model and an algorithm for the considered binomial compression of equilibrium combinations; -construct a mathematical model and an algorithm for the considered binomial compression of binary sequences of general form.

1. Justification of compression expediency
The proposed binomial compression methods are based on binary (n,k)-binomial numbering systems and binary binomial numbers generated by these systems (n and k above are parameters of the used set of numbers).
Binary (n,k)-binomial numbering system is characterized by: 1) a binomial numerical function that allows one to determine the decimal quantitative equivalent F j of a binary binomial number X j = x 1 x 2 ...x i ...x r [11,12]: where X j ÎX[n,k], x i Î{0,1}, r < n, j C n k = 1 2 , ,..., ; q i is the sum of unit digits x i from the first to the (i-1)-th position, inclusive: 2) systems of code-forming constraints that allow one to generate binary (n,k)-binomial numbers X j = x 1 x 2 ...
where q and l are the numbers of ones and zeros in the binary binomial number X j . Binomial numerical function (1), being a numbering function, allows one to associate binary (n,k)-binomial numbers X j with their decimal equivalents F j or binary numbers D j = BinF j D D n k C  1 : , , define, respectively, binomial compression and restoration based on binary binomial numbers where 0 ≤ k ≤ n [12].
Justification of expediency of developing a method for compression of binary equilibrium combinations Y j ÎY[n,k] based on the binomial numerical function (1) includes the following statements: 1) length of numbers D j satisfies the inequalities |D j | < n and |D j | < L ср where L ср is the average length of binary (n,k)-binomial numbers [10] which provides a compression ratio of more than one; 2) functionality of compliances H D n k X n k , , (D j ,X j )ÎH and S X n k D n k , , (X j ,D j )ÎS between sets D[n,k] of binary numbers D j and X[n,k] of binary binomial numbers X j based on the theorems of unambiguity of binomial numbers [10,12]. This leads to bijective binomial mappings of the form ψ -1 :D[n,k]→X[n,k] and ψ:X[n,k]→D[n,k] and ensures one-to-oneness of encoding and decoding, respectively; 3) uniformity of binary numbers D j enabling significant reduction of hardware and software costs in the construction of devices and software modules for compression and restoration of equilibrium combinations Y j based on the binomial numerical function (1).

2. The method of compression of equilibrium combinations based on the binomial numerical function
Realization of binomial mapping of the form ψ: is represented by compression: of equilibrium combinations Y j ÎY[n,k] based on binary numbers D j ÎD[n,k] which are calculated using the function (1). This type of compression will also be called binomial nu-merical compression. Thus, we are dealing with a complex function f f e b = ψ  : In turn, the realization of the binomial mapping ψ -1 : of initial equilibrium combinations Y j ÎY[n,k] based on binary numbers D j ÎD[n,k] which are calculated using the function (1). This type of restoration will also be called binomial numbering restoration. Thus, we are dealing with a complex inverse function f f , , D n k X n k which are determined by the numerical function (1) and systems of code-forming constraints (2). In other words, when compressing and recovering based on the binomial numerical function (1), binary binomial numbers X j are used when passing from equilibrium combinations Y j to binary numbers D j and vice versa. This significantly distinguishes the proposed compression method from the numbering systems considered in [6,14].
The following Theorem 1 gives properties of compressive mapping f e and the method of its practical implementation («/» is the decatenation symbol).
Theorem 1. Any binary sequence Y j = y 1 y 2 ...y i ...y n , Y j ÎY[n,k], j C n k = 1, composed of n bits y i , the sum of values of which is equal to k, can be associated with a unique positive integer D j ÎD[n,k] using mapping f e in two steps: 1) transition from the sequence Y j = y 1 y 2 ...y i ...y n to the binary (n,k)-binomial number X j = x 1 x 2 ...x i ...x r , X j ÎX[n,k], r < n using the function X j = f b (Y j ) in the form: 2) calculation of the number D j = Bin(decX j ) of the binary (n,k)-binomial number X j in accordance with the numerical function (1).
Proof of Theorem 1 for the first stage of transformations is made proceeding from compliance of X j to the systems of code-forming constraints (2) and the uniqueness of X j taking into account the fact that the decatenation operation does not change values of the digits x i . At the second stage of transformations to prove Theorem 1, the sequential division of decX j by C n i k q i − − is used to obtain the quotient 0 ≤ δ i ≤ 1 and substantiate uniqueness of decomposition (1).
Reflecting the form of a complex function f f e b = ψ  , the model of f e compression process includes the first two stages related to f b compression based on binary binomial numbers [12].
Thus, using Theorem 1, modeling of the f e compression process of binary equilibrium combinations Y j = y 1 y 2 ...y i ...y n based on the binomial numerical function (1) consists of the following stages.
Stage 1. The value of the last byte y n is determined in the n-digit equilibrium combination Y j = y 1 y 2 ...y i ...y n having k of ones.
Stage 2. If y n = 0, then that is, all zero bits are discarded from the combination Y j = y 1 y 2 ...y i ...y n-1 0 starting with y n = 0 until the first binary one y r = 1appears which will represent a value of the last bit x r = y r = 1 of the sought (n,k)-binomial number that is, all unit digits are discarded from the combination Y j = y 1 y 2 ...y i ...y n-1 1 starting from y n = 1 until first binary zero appears y r = 0 which will represent a value of the last bit x r = y r = 0 the sought (n,k)-binomial number In this case, in both cases, values of the remaining bits remain unchanged: Stage 3. Quantitative equivalent decX j of the binary (n,k)-binomial number X j = x 1 x 2 ...x i ...x r is calculated in accordance with the numerical function (1).
Stage 4. The quantitative equivalent of decX j is converted to its binary form D j = Bin(decX j ) to obtain thereby the sought number D j ÎD[n,k], a compressed image of the original n-digit sequence Y j = y 1 y 2 ...y i ...y n .
The following Theorem 2 gives properties of f e −1 mapping of restoration and the way of its practical implementation («++» is the symbol of the concatenation operation).
Theorem 2. Any binary number D j ÎD[n,k] can be put in one-to-one compliance with a unique binary sequence  ψ reflects the fact that the steps related to the f b −1 restoration based on binary binomial numbers [12] will be included in the model of the f e When applying the binomial numeric function (1), there is a more general case when k can take any values from the range 0 ≤ k ≤ n and the compressed array is a set Binary numbers D j are uniform in length only for a constant value of k. Hence, to unambiguously restore A j Î{0,1} n , from the D j numbers, it will be necessary to additionally use the k value expressed in binary form Bin k. This means that the Bin k should be appended to the resulting binary numbers D j . This approach is justified if 0 < k < n. For the case when (k = 0)∨(k = n), to unambiguously restore the sequence of zeros and ones, it is sufficient to transmit and store only the binary word Bin k. Therefore, additional coding is introduced in the f k form: Also, in the general case for 0 ≤ k ≤ n when compressing A j , it is necessary to use the function f w : A→M, Further, if the obtained value of k satisfies the inequality 0 < k < n, then to compress the equilibrium combination Y j complying with A j , f e coding based on a binomial numerical function is used. In this case, Bin k is added to the compressed combinations for unambiguous restoration, i. e., f k is additionally encoded. If the k value satisfies the system of equalities (k = 0)∨(k = n), then the encoded resulting combination will consist only of Bin k, i. e., only the f k encoding method is used.
Thus, the general case of compression when 0 ≤ k ≤ n, is represented by a mapping in the form f eg : A→Z which is set by the corresponding function Z j = f eg (A j ) where A j = a 1 a 2 ...a i ...a n , A j ÎA = {0,1} n , Z j = (Bin k,D j ) or Z j = Bin k, Z j ÎZ, j n = 1 2 , . Theorem 3 below gives properties of f eg mapping and the way of its realization.
Theorem 3. Any binary sequence A j = a 1 a 2 ...a i ...a n , A j ÎA = = {0,1} n , j n = 1 2 , , can be put into a one-to-one compliance with a binary combination Z j ÎZ of the following form: 2) otherwise, if (k = 0)∨(k = n), then Z j = Bin k. Proof of Theorem 3 is based on the fact that for the case of 0<k<n there cannot be two or more numbers of k ones and the numbers D j (Theorems 1, 2) for the compressible sequence A j and for the case of (k = 0)∨(k = n), the validity of the theorem is confirmed by the uniqueness of A j in which all digits are either zeros or ones.
The mapping f eg : A→Z is bijective since each element of A j has a unique image and each element of Z j is the only preimage for all A j ÎA and Z j ÎZ.
Thus, the mapping f eg : A→Z which is given by a complex function in the form: is called a generalized method of compression based on a binomial numerical function (1) or a generalized binomialnumbering compression. Obviously, the previously considered compression f e is a composite component of the generalized compression f eg based on the binomial numerical function (1). Taking into account that f f e b = ψ  , binary (n,k)-binomial numbers are intermediate code objects in the process of information transformations during f eg compression. Such relationships between f b , f e and f eg simplify practical implementation when they are used together, expand their functionality, and make it possible to use an adaptive approach to binary compression of information using binary binomial numbering systems.
In turn, the inverse mapping f Z A eg − → 1 : , which is given by the inverse complex function: is a recovery taking into account the existing value k of the initial binary sequences A j .
In the case when 0 < k < n, A j is recovered on the basis of Bin k and binary numbers D j ÎD[n,k] using binomial numbers X j ÎX[n,k] when passing from the number D j to the equilibrium combination Y j . In the case of k = 0 or k = n, A j is recovered based on Bin k by generating n zeros or ones, respectively. This type of recovery will be called generalized recovery based on a binomial numerical function.
Mappings f e and f e −1 are constituent elements of the generalized compression and recovery methods f eg and f eg −1 , respectively. Hence, the methods of constructing complex functions (6) and (7) will be in many respects similar to the methods of implementing the methods of compression and restoration of equilibrium combinations (Theorems 1 and 2) when 0 < k < n. Methods of implementing the functions of f eg compression (6) and f eg −1 recovery (7)  Stage 4. If the number k satisfies the system of equalities (k = 0)∨(k = n), then (k = 0)∨(k = n) will be the resulting combination. Otherwise, the available n value and calculated k-value are parameters of the binary (n,k)-binomial numbering system and the transition to the subsequent stages is made to implement coding f k (f e (Y j )) = Z j , Z j ÎZ e. Stage 5. The number m of bits is calculated for binary representation of the number D j ÎD[n,k] corresponding to Y j ÎY[n,k]: m C n k = é  ù  log . 2 Next, Stages 6-9 of the model of the f eg compression process correspond, respectively, to Stages 1-4 of the model of the f e compression process.
Stage 10. Concatenation of the binary value Bin k and the number D j , i. e., encoding of the form f k (D j ) = Z j is carried out for the case 0 < k < n: Z j = Bin k+D j , thereby obtaining the resulting combination Z j ÎZ o .

1. Estimation of compression of equilibrium combinations based on the binomial numerical function
The results of compression f e :Y [8,2]→D [8,2] for some Y j ÎY [8,2] using the corresponding intermediate X j ÎX [8,2] are shown in Table 1. For a complete set Y [8,2] when average length L 8,2 of binomial numbers X j ÎX [8,2] is 5.71 bits (according to [10]) and the number of binary digits to represent the number is D j ÎD [8,2] is L C n = é  ù  = log .  Table 1, the compression ratio of 8-bit equilibrium combinations Y j ÎY [8,2] when mapping f e will be K e = n/L n = 8/5 = 1.6. It is also obvious that if the degree of binomial compression should be increased, it is necessary to switch from f b compression based on binomial numbers to f f e b = ψ  compression based on the binomial numerical function (1).
In general terms, the ratio K e of f e compression is a ratio K n C e n k = é  ù  log , 2 where the denominator is the number of digits to represent the binary number D j . Table 2 demonstrates a variation of the coefficient K e depending on value k of ones in the 128-bit equilibrium combinations Y j ÎY[128,k] having a length of n = 128. Table 1 Compliance of some Y j ÎY [8,2], X j ÎX [8,2] and their numbers with D j ÎD [8,2] No. Equilibrium code Y [8,2] Set X [8,2]   As seen from Table 2, compression of equilibrium combinations Y j ÎY[128,k] based on the binomial numerical function (1) is observed for any fixed value 1 ≤ k ≤ 127 of binary ones (because of the uniqueness of corresponding combinations, the values k = 0 and k = 128 are not considered). In this case, a minimum value of the ratio K e of compression f e is 1.02, and the maximum is 18.29.

2. Estimation of compression of binary sequences of general form based on the binomial numerical function
When applying the generalized binomial f eg compression based on the binomial numerical function (1), one should additionally calculate the number k which is now already a variable and concatenate Bin k and the resulting number D j .
On the one hand, the use of Bin k slightly reduces the resulting compression ratio but, on the other hand, it significantly expands the scope of f eg compression since it can be used for ordinary information arrays. It should also be noted that the Bin k/n ratio tends to zero at n→µ which means a decrease in Bin k influence on the compression ratio with an increase in length n of the compressed sequences A j .
Some results of the considered mapping f eg :A→Z and, therefore, f Z A eg − → 1 : for n = 24 where A j ÎA = {0,1} 24 , j = 1 2 24 , , are given in Table 2 Table. For each sequence A j , Table 3 shows both corresponding binary (24,k)-binomial number X j used in the transition to the number and the resulting number D j itself.
In a general form, ratio K eg of compression f eg is the ratio K n n C e n k = + ( ) where the denominator is the sum of the number of digits to represent the binary Bin k and the number D j . Table 4 shows the variation of the coefficient K eg depending on the value k of ones in the 128-bit binary sequences A j ÎA = {0,1} 128 , j = 1 2 128 , . The number k of ones can have any variable value in the range of 0 ≤ k ≤ 128.
As seen from Table 4, compression of binary sequences of general form A j based on the binomial numerical function (1) is observed at values of 0 ≤ k ≤ 48 and 80 ≤ k ≤ 128 binary ones. In these ranges of k variation, a minimum value of ratio K eg of f eg compression is 1.01, and the maximum one is 16. In the case of 48 < k < 80, the f eg compression ratio will have the value of K eg = 0.98 or K eg = 0.96 (for k = 64), i. e., less than one which is the consequence of including Bin k in the resulting combination for unambiguous recovery of A j .

Discussion of results obtained in the study of compression methods based on the binomial numerical function
The mathematical model (4) of the process of compression f e of equilibrium combinations Y j contains the function f b of obtaining binary binomial numbers X j from Y j . This makes it possible to significantly reduce the number of computations when determining the binary number D j according to (1), since length r of the numbers X j is on average significantly less than the length n of combinations Y j . Therefore, there is no need to use (n-r) weight coefficients C n i k q i − − , as it is done in [6,7]. As Table 2 shows, information redundancy in equilibrium combinations can be eliminated from 1.02 to 18.29 times. Accordingly, the speed of information transmitted through the communication channel can be increased the same number of times.
The  Table 3 Compliance of f eg between some binary A j and Z j for n = 24 The mathematical model (6) of the process of f eg compression of binary sequences of a general form A j contains f e compression coding as a composite function, and therefore also uses binary binomial numbers X j . Accordingly, this leads to a reduction in computational costs when searching for the D j number, as it does for f e mapping. Inclusion of function f k of the concatenation of binary values Bin k and the D j number in model (6) leads to a decrease in compression ratio though this decrease is insignificant. As Table 4 shows, elimination of information redundancy in binary sequences of a general form is from 1.01 to 16 times for n = 128. Compared to f e compression, the use of Bin k leads to a decrease in compression ratio by approximately 6-16%. However, for n→µ, the influence of Bin k on the compression ratio f eg becomes more and more insignificant since the number of élog 2 (n+1)ù bits for the Bin k representation grows much slower than that for n.
The formulated Theorem 3 determines the unambiguity of f eg coding and is the basis for constructing a model of the process of the corresponding compression. The f eg modeling steps shown include the model of f e compression which makes it easier to synthesize them together. Additional operations of calculating the number k of ones and concatenating the Bin k to the compressed image D j feature simplicity and minimal hardware and software costs in practical implementation.
As indicated in [14], the compression coefficient for the methods based on the binomial numerical function at n→µ asymptotically tends to the limiting value, i. e., reciprocal of the entropy H(p) where p is the probability of appearance of one.
Not only the compression ratio but also the time of encoding and decoding of binary sequences have a significant impact on compression efficiency from the point of view of performance improvement [1][2][3].
Inclusion of the function f w in model (6) for calculating k of ones increases the time of f eg compression by a maximum of én/2ù computation cycles [12]. But this is significantly less than the time required to get the actual number D j .
The methods of f e and f eg compression are based on the numerical function (1) and calculation of binomial coefficients C n i k q i − − , that play the role of weights in determining the numbers D j of compressed binary sequences. The degree of binomial numbering compression is in direct proportional dependence on the length n and the number k of ones of a binary combination. However, an increase in n and k leads to a significant increase in computational costs when determining weights C n i k q i − − of the numerical function (1) which, in turn, leads to a noticeable increase in costs for hardware, software, and/or timetable [15,16].
For the practical implementation of f e and f eg compression methods, it is proposed to use the tabular method of calculating binary numbers D j in which all weight coefficients C n i k q i − − pre-calculated for parameters n and k will be used as table elements. The timetable here will be determined only by adding q values of C n i k q i − − . Then the hardware costs will consist of the number of memory cells required to store all the weight coefficients, and the time spent from the time of adding the weight coefficients and the timetable consists of calculating the number k of ones which is limited by the value k ≤ én/2ù.
It should also be noted that knowledge of k in the case of compression f e of equilibrium combinations and the calculated value of k in the case of generalized binomial compression f eg make it possible to control errors. If, other values of k are obtained that differ from the original ones when transmitting compressed images through communication channels and their subsequent restoration, this indicates erroneousness of the resulting sequences. The obtained feature of compression methods based on the binomial numerical function (1) is an additional way to ensure the specified validity of data transmission and the level of fault tolerance of the information systems.
Thus, in contrast to [11][12][13], mathematical models of f e and f eg compression were obtained in this study based on the binomial numerical function (1) and stages of modeling the corresponding processes were also presented.
In contrast to [4,5,14], the advantage of this study consists in the abstraction of binary binomial numbers in the structure of compressible binary sequences. This makes it possible to reduce the time spent on calculating the number D j according to the binomial numerical function (1) and introduce control of errors that may arise during the transfer of compressed images and their subsequent recovery.
As a disadvantage of the above study, the absence of models of processes of recovery of f e −1 and f eg −1 and initial combinations based on the binomial numerical function (1) can be pointed out. This is explained just by the limited volume of this paper.
Further development of the proposed compressive coding methods is seen in developing algorithms and devices for practical binomial compression as well as estimating the compression degree and time depending on probabilistic characteristics of binary information sources.
The exemplary area of application of the developed compression methods based on the binomial numeric function includes the following: -data transmission systems using equilibrium or quasiequilibrium codes [17]; -database management systems storing information sequences, the sum of the elements in which is limited or equal to a set limit [6,13]; -systems for automation of scientific experiments in which their statistical characteristics are unknown at the preliminary stage of data processing; -information and control systems using binomial sets of numbers for solving specialized problems [12].

Conclusions
1. A mathematical model of the method of compressing equilibrium combinations based on a binomial numerical function has been constructed. This model is based on Theorems 1 and 2 revealing the procedure of the method implementation and makes it possible to implement sufficiently effective models of the processes of binomial numbering compression and restoration of equilibrium combinations. In this case, binary binomial numbers are used as intermediate numbers in the transition to binary numbers.
2. A mathematical model of the generalized compression method based on a binomial numerical function for binary sequences with a variable number of ones has been constructed. Theorem 3 revealing the procedure of implementing this method and enabling the implementation of sufficiently effective models of processes of binomial generalized compression and restoration of information sequences of a general form is the basis of the constructed mathematical model. In this case, the numerical compression models of equili brium combinations are an integral part of the generalized compression models based on the binomial numerical function.