The purpose of this work is to develop and comprehensively study effective methods of using vector algebra tools to solve practical problems of information coding with the integration of modern computing technologies aimed at increasing the productivity, reliability and adaptability of the corresponding systems. In the light of the rapid development of information technologies and the growth of the volume of data that needs protection, it becomes relevant to design algorithms, which would provide not only security, but also the best speed in real time. This article aims not only to substantiate theoretically such methods, but also to realize them in practice on software, namely in the Python programming language, which is a widely used platform for scientific and technical computing. Particular attention is given to development of algorithms capable of working with matrices of various dimensions, including 4 x 4 and 5 x 5, in finite fields, such as GF (17) that provides resistance to noise and errors in the data transmission process. Thus, the goal of the article includes the theoretical analysis and experimental verification of the proposed approaches with special attention to their practical worth to modern coding and cryptography systems.
A unique integrated approach to the implementation of vector methods in the information encoding processes is proposed, based on optimized algorithms of matrix operations and their software implementation in the Python environment, which is an innovative contribution to the development of relevant technologies. Unlike traditional methods that usually make straightforward linear transformations, it incorporates modern notions of vector spaces and matrix algebra with modifications for finite field conditions. A special novelty is the use of the Gauss-Jordan method on calculating the inverse matrix in GF (17), which makes it possible to avoid rounding errors typical for numerical methods in real numbers.
Within the framework of the research, mathematical model of vector coding was developed, software tools were developed to automate the coding and decoding processes, in particular, based on the language Python using the library NumPy. Experimental studies were conducted, which testify the high efficiency of the proposed methods for matrices of dimensions 4 x 4 and 5 x 5 in the finite field GF (17). The algorithm, which is implemented by the Gauss-Jordan method, shows the correctness of the recovery of input vectors even in the presence of error vectors, which indicates the system's resistance to noise.
The process of using vector algebra tools allows a significant improvement in information coding system efficiency, an optimal balance between execution speed and reliability of processing the received data in the modern IT sector. The results obtained highlight the fact that the combination of matrix algebra with vector operations in finite fields, such as GF (17) enables you to develop noise-resistant systems which can be tailored to different dimensions and to different operating conditions. Experimental data show high accuracy of information recovery even at error, which is critical for telecommunication systems and cryptography protocols. At the same time, further research can be conducted towards the optimization of computational complexity e.g. using parallel computing or integration with hardware accelerators, e.g. GPUs. Therefore, the proposed work has good potential for innovative solutions in the field of protection of data and optimization of coding processes of computer systems, which opens new horizons for scientific and technical research in this direction.
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