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Full Non-Binary Constant-Weight Codes

Abstract

This paper presents the research results of mixed base number systems using binomial representation of numbers. It also shows the investigated non-linear coding techniques with full constant-weight codes, which are based on the binomial numeration. These codes can be used in a variety of computer applications: in code-based cryptosystems; to detect errors in asymmetric communication channels, etc. We propose a new numeral system. It combines the features of positional and binomial numeration. We suggest a technique of full non-binary constant-weight coding based on a generalized binomial-positional representation. It allows us to generalize the known approach to the non-binary case and practically implement computational algorithms for generating full set of non-binary constant-weight sequences. There are the analytical relations that relate the positional and binomial representations of numbers. The article provides several examples, which clear up the usefulness and the constructiveness of the proposed approach, and simplify the perception and understanding of results. We also consider some aspects of the potential use of the proposed numeral system. In particular, the article discusses asymmetric communication channels (the model of a binary Z-channel and its generalization to the non-binary model), and shows the advantage of constant-weight codes for error detection.

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Acknowledgements

This work was supported in part by the National Research Foundation of Ukraine under Grant 2020.01/0351.

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Correspondence to Alexandr Kuznetsov.

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Kuznetsov, A., Kryvinska, N., Kiian, A. et al. Full Non-Binary Constant-Weight Codes. SN COMPUT. SCI. 2, 337 (2021). https://doi.org/10.1007/s42979-021-00739-w

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Keywords

  • Full constant-weight codes
  • Mixed base number systems
  • Binomial count
  • Z-channel
  • Code-based cryptosystems