## 5 Conclusions  
This paper proposes a lossless adaptive binomial data compression method that is efficient for hardware implementation and based on binomial numbers that allow for compressing binary messages of code length \(n\) while controlling their correctness and, if necessary, changing the compression ratio depending on the requirements for it in terms of speed and noise immunity. The presence in the compression method of a separate source of information that generates keys \(k\) makes it possible to use it for the transmission or storage of confidential information and, if integrated into the information protection system, to increase its efficiency. The compression coefficients for different lengths of sequences and the number of \(k\) units contained in them are studied. For these adaptations, a two-sided restriction on the value of \(k\) is introduced. The excess on one or the other side prohibits the compression of binary sequences. The value of \(k\) can be either large or small by the symmetry properties of binomial coefficients. Changing these values allows, by changing the compression ratio, to reduce or increase the speed of information compression. Using noise-immune binomial circuits makes the proposed method convenient for hardware implementation and protects information from unauthorized access. The practical application of the proposed adaptive binomial compression method is most effective in mobile communication systems operating under interference conditions and compressing texts, graphics, and images. In this case, the longer the length of the compressed binary block, the more efficient the compression. The method, under certain conditions, can surpass the known compression methods in terms of speed, compression ratio, and noise immunity. The peculiarity of the method is its adaptability, which means the possibility of finding the maximum efficiency under the conditions of changing probabilities of generated binary sequences. It can give a gain in performance and, at the same time loss in the compression ratio and vice versa. The disadvantage of the method is its Bernoulli sources of information and the presence of increased complexity associated with the need to introduce a tracking system for compressible sequences. It allows for the exclusion of inefficient sequences from the compression procedure.  
Conceptualization, S.M. and O.B.; methodology, S.M. and O.B.; software, S.M.; validation, S.M., O.B. and V.B.; formal analysis, S.M.; investigation, S.M.; resources, S.M. and O.B.; data curation, S.M. and O.B.; writing--original draft preparation, S.M., O.B. and V.B.; writing--review and editing, S.M., O.B., T.S. and S.S.; visualization, S.M.; supervision, V.B.; project administration, V.B.; funding acquisition, S.M. and V.B. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the European Regional Development Fund within Activity 1.1.1.2 \"Postdoctoral Research Aid\" of the Specific Aid Objective 1.1.1 \"To increase the research and innovative capacity of scientific institutions of Latvia and the ability to attract external financing, investing in human resources and infrastructure\" of Operational Programme \"Growth and Employment\" (No. 1.1.1.2/VIAA/3/19/421).  
Not applicable.  
The authors declare no conflict of interest.