The mΘ Protocol F5 and Hamming mΘ Codes

Author: Dr. Pemha Binyam Gabriel Abstract: Chrysippian Ө-valent modal logic is a mathematical logic with several truth values that challenge...

Author: Dr. Pemha Binyam Gabriel

Abstract:

Chrysippian Ө-valent modal logic is a mathematical logic with several truth values that challenge Boolean logic. In 1930, Lukasiewicz constructed similar systems for a countable infinite chain of intermediate truth values between 0 and 1. However, the formulation, the algebraization of chrysippian Ө-valent modal logic is due to Wajsberg in 1951. It should be noted that Ө-valent modal logic does not verify the principle of the excluded middle, which led to comparing intuitionist logic to Ө-valent modal logic.

Later, in 1982, Fidèle Ayissi laid the foundations of chrysippian Ө-valent mathematics. From then on, we no longer speak only of Ө-valent modal logic but of Ө-valent mathematics. Ө-valent mathematics is based on chrysippian Ө-valent modal logic with some similarities with the fuzzy mathematics of Zaddeh.

Just as in classical mathematics, we talk about algebraic structures (set, group, ring, field, …); at the same time, in Ө-valent mathematics, we also talk about Ө-valent modal algebraic structures (mӨ set, mӨ group, mӨ ring, mӨ field, …). However, these Ө-valent modal algebraic structures are faithful to a certain modal compatibility. For example, let (E, F α ) be a mӨ set, for all x and y in E, x= Ө y if and only if for all α in I ⁎, F α (x)= F α (y).

As if to say that it is a question of using a chrysippian Ө-valent modal logic to construct intrinsic mathematics leading to various applications, such as coding theory, number theory, algebraic combinatorics, etc.…. Remember that the application of modal logic in analysis also gives valuable results. In my research work, i attempt an Ө-valent modal approach on finite sets, in discrete algebra. We work on mӨ codes, cryptography based on mӨ codes, steganography and mӨ codes.

Biography:

1. J.A. Tsimi and G. Pemha, 2021, A mƟ spectrum of Reed-Muller codes, Journal of Discrete Mathematical Sciences and Cryptography (JDMSC), doi: 10.1080/09720529.2020.1814489

2. J.A. Tsimi and G. Pemha, 2021, On a Decoding algorithm of mƟ Reed-Muller codes, Journal of Discrete Mathematical Sciences and Cryptography (JDMSC), Volume 26, Issue 2, pp: 341-358, doi: 10.1080/09720529.2021.1920189

3.  J.A. Tsimi and G. Pemha, 2021, On the Generalized modal Ɵ-valent Reed-Muller codes, Journal of Information and Optimization Sciences (JIOS), Vol 42, 2021, Issue 8, 1885-1906. doi: 10.1080/02522667.2021.1961977

4. Jean Armand TSIMI, Pemha Binyam, Ketchandjeu Armand, On Modal Θ-valent steganographic protocols, Journal of Mathematical Sciences: Advances and Applications, Volume 71, 2022, pp. 5-46.

5. Gabriel Cedric Pemha Binyam, Laurence Um Emilie, Yves Jonathan Ndje. The mΘ Quadratic Character in the mΘ Set ZnZ. Mathematics and Computer Science. Vol. 8, No. 1, 2023, pp. 11-18. doi: 10.11648/j.mcs.20230801.12

6. PEMHA BINYAM Gabriel Cédric, Quasi-Cyclic Codes Over Finite Chain 𝒎Θ Pseudo Field F(𝒑𝒌Z, 1), Global Journal of Science Frontier Research: F Mathematics and Decision Sciences, Volume 23, Issue 3, Version 1.0, 2023, pp. 85-101.

7. PEMHA BINYAM Gabriel Cédric, The mӨ protocol F5 and Hamming mӨ codes, London Journal of Research in Science: Natural and formal, volume 23, issue 13, 2023, pp. 21-36

Article link: https://journalspress.com/LJRS_Volume23/The-m%CE%98-Protocol-F5-and-Hamming-m%CE%98-Codes.pdf?