Evaluation of multidimensional entropy on short strings of biometric codes with dependent bits

The article considers a problem of evaluating output codes entropy at the neural network "biometrics-code" transformer of the analogue (continuous) biometric data into output digital code. For the codes with independent bits (white noise) the entropy matches the length of the binary code. The problem becomes sophisticated if the code bits of the string under investigation turn out to be dependent (correlated). Evaluation of the dependent bit codes' entropy is carried out during the analysis of languages (artificial and natural), as well as during the testing of biometric codes, obtained in the output of neural network "biometrics-code" transformers. Investigating the entropy of emergence of 8-bit codes of single letters in the texts of Russian and Kazakh languages, the researchers have encountered the bit dependence in the 8-bit code string. The same dependence emerges during the investigation of 16-bit codes of letter pairs, 32-bit codes of letter quadruples,. ., 256-bit codes of 32-letter groups of text in Russian and Kazakh languages. Attempts to determine the code entropy of the natural language by means of the classic technique (through estimating the probability of either code emergence) appear to be a problem of exponentially increasing complexity. The importance of the given problem emphasizes even more when testing the neural network human biometry transformers of personal keys into 256-bit codes. The objective of the article is to evaluate the entropy of various length codes with dependent bits. The authors consider the dependence (correlation) of the states of the code within its length, as well as the correlation (dependence) of the states of the same bits of different codes in the string under investigation. The article shows that the output multidimensional entropy significantly depends on the dual correlations of output code bits. In the course of statistical analysis the researchers has built a nomogram of correlation values of the entropy of different dimensions with an averaged value of dual correlations module. The given nomograms display that data independence (noncorrelatedness) requirements considerably vary with an increase of problem dimensionality. With low amount of outputs up to 16, the independence (noncorrelatedness) requirements remain low. They are negligible. For "biometrics-code" transformers with high number of outputs n = 128; 256; 512; 1024 it is necessary to make severe demands to the independence (noncorrelatedness) of output code bits (average value of correlation coefficient modules should be no less than 0.15, which corresponds to the standard CTP 52633.0-2006).