## 2 General Analysis of the Binomial Adaptive Compression Method  
### The Binomial Decomposition of Information Sources  
The lossless binomial adaptive compression of binary sequences of length \(n\) proposed by the authors is based on the binomial information compression they developed based on the theorem on the decomposition of Bernoulli sources of information [20]. The theorem proves that the Bernoulli source of probabilistic binary sequences \(A^{*}\) decomposes into two sources, one of which is a combinatorial source \(A\) with conditional entropy \(H(A/B)=\sum\limits_{k=1}^{n}\hat{P}_{k}\log_{2}C_{n}^{k}\), generating equally probable binary sequences of different code lengths, and another probabilistic source of information \(B\) with entropy \(H(B)=-\sum\limits_{k=1}^{n}\hat{P}_{k}\log_{2}\hat{P}_{k}\), generating quantities of \(k\) number of units \(0\), \(1\), \(\ldots\), \(n\) in these sequences.  
Probabilities \(\hat{P}_{k}\) determine the probabilities of occurrence of these quantities. It is proved that the sum of the entropies (entropy of the union) of the sources \(A\) and \(B\) are equal to the entropy \(H(A^{*})=-\sum\limits_{j=1}^{2^{n}}p_{j}\log_{2}p_{j}\) original source of binary sequences \(A^{*}\), where \(p_{j}\)--source generation probability \(A^{*}\) of \(j\)th binary code combination.  
The Bernoulli information source entropy equation is decomposed into two entropies. They provide the basis for new compression methods, one of which is discussed in this article. The entropy decomposition equations show that the Shannon-Fano and Huffman statistical methods of optimal coding can be effectively replaced by numerical coding methods that do not require preliminary statistical tests for their implementation. In this case, the compression efficiency in practice is not less than in statistical methods of optimal coding. With that compression occurring with zero errors in statistics, it is even higher. This idea underlies the compression method [20] previously presented by the authors, which was carried out using binomial numbers [21; 22; 23]. However, compression occurs for different binary sequences with different efficiencies. The difference here depends on the probability distribution of generating compressible sequences.  
To eliminate the time spent on compression, it was proposed to abandon the compression of inefficiently compressible sequences. As a result, practically without reducing the compression efficiency, it was possible to reduce the compression time several times while increasing its noise immunity. However, this process requires the authors' dynamic control system called adaptive coding. The solution to the problem of improving efficiency proposed by the authors formed the basis of this paper. The binomial compression method itself, without adaptation, has long been studied by the authors and has shown its practical performance. This adaptive compression method development has been proposed for the first time for publication. The authors also developed a program that confirms the practical performance of the adaptive compression method. In any case, the efficiency of the proposed adaptive compression method is higher than without adaptation.