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In 1978, McEliece built the first public key cryptosystem based on error-correcting codes. This cryptosystem based on Goppa codes is considered promising and cryptographically stable, taking into account quantum computing. At the same time, effective attacks on the secret keys of this cryptosystem have not yet been found. In the paper, algorithms for decoding Goppa codes in the case of errors and erasures are investigated. Four decoding algorithms based on the algorithms for Reed–Solomon codes proposed by Gao, Berlekamp and Massey, Sugiyama, and others are given. The first two algorithms are based on Gao algorithm and related to syndrome-free decoding algorithms, the rest are related to syndrome decoding algorithms. Moreover, any of these algorithms is also applicable for the case of a communication channel with errors only. Examples of decoding separable Goppa codes using these algorithms are also given.

The relevance of the topic is determined by the need to increase the speed of data transmission in satellite telecommunication systems through the use of noise-proof coding. The speed of data transmission in a radio channel is actually determined by the level of interference present due to the fact that the radio transmission medium is a public environment and other electronic devices operate in it. The purpose of the article is to develop an algorithm for encoding and decoding the input signal using a multithreshold decoder (MTD) of rigid solutions for subscriber terminals operating with low-orbit spacecraft (NCA), and its comparison with a conventional PC decoder «Messi». The comparison allows us to conclude that the application of the developed algorithm is justified in systems where more stringent requirements are imposed on the probability of an error in the communication channel, while the «Messi» decoder is more appropriate to use in the presence of restrictions on computational complexity.

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A training variant of Reed-Solomon codes, which greatly simplifies the process of synthesis of codewords and detection-fixes for corrupted data files. The algorithm is based on the transfer of variables and operations, in which the synthesis and analysis of codes from the original space into isomorphic space of image. As a result, the proposed replacement of data space computational process is reduced to the basic operations in a simple and modular Galois field arithmetic on integer operands, easily implemented by means of computer technology.