## 1 Introduction
Accordingly, when transmitting the values \(m\) and \(n\), instead of compressing the binary sequence, its lengthening will be observed due to redundant coding since the transmission of the length of the compressed number m is also added to the transmission of the length of the binary number for the number of units \(k\). In all other cases of compression of binary sequences most often encountered in practice, the inequality \(n>k>0\) takes place. This inequality allows the compression to be adapted to its required speed. A restriction is introduced on the value of \(k\), the excess of which prohibits the compression of binary sequences. Changing this value reduces or increases the information transfer rate due to the compression ratio. The possibility of such regulation is the effect of the adaptability of the proposed compression method. When restrictions generally prohibit the compression of a number of binary sequences, it is possible to increase their noise immunity by transmitting only the \(k\) value to the receiver during compression. The compression effect of the proposed method depends on the probabilities of the distribution of the number of units in the compressed binary code sequences. The greater the likelihood of sequences with a number of ones different from \(k=n/2\), the greater this effect will be. However, it cannot exceed the bounds given by Shannon's equation for the entropy of binary sequences. Therefore, this method is comparable in terms of compression efficiency with existing lossless information compression methods. Its main advantage over them is the ability to change the compression efficiency depending on the requirements for its performance. Another essential property of this binomial compression method is it increases the noise immunity of a part of transmitted binary messages. In addition, using the binomial number system in this compression method makes it possible to effectively implement it in hardware while increasing data compressions speed, reliability, and noise immunity. A distinctive feature of the proposed method for compressing binary sequences of finite length \(n\) is their numbering using the binomial number system (BNS). In addition, the method allows the compression ratio \(K\) to be adapted to its required speed and error detection. Adaptation manifests itself in the fact that for each compressed sequence, the compression coefficient \(K\) is preliminarily determined, and, depending on its value and the requirements for speed and error detection, a decision is made whether or not to perform the compression operation. As a result, only those sequences are compressed for which the compression coefficients \(K\) are more significant than the specified value. The remaining possible sequences are transmitted or stored in their original form. Using the binomial number system and binomial numbers in the method makes it possible to implement it in hardware. The availability of keys for each compressed sequence can be used in systems to protect information from unauthorized access. The rationale for developing methods for compressing information based on binary binomial numbers generated by binomial number systems is:  
(1) The non-uniformity of binomial numbers whose length \(r\) is less than the length \(n\) of the original compressible code combinations, which provides a compression coefficient more significant than one;  
(2) The functionality of correspondences between sets of binary binomial numbers and combination codes and provides one-to-one coding and decoding;  
(3) The prefix of binary binomial numbers, which allows, without additional hardware (and/or software) and time costs for separators, compression coding, and restoration of binary sequences;  
(4) Prevalence of code combinations based on binomial numbers (for example, equilibrium and quasi-equilibrium codes, combinations with restrictions on mutual arrangement of zeros and units, etc.) for data presentation in information-control systems.