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Fibonacci matrices, a generalization of the “Cassini formula”, and a new coding theory

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  • Stakhov, A.P.

Abstract

We consider a new class of square Fibonacci (p+1)×(p+1)-matrices, which are based on the Fibonacci p-numbers (p=0,1,2,3,…), with a determinant equal to +1 or −1. This unique property leads to a generalization of the “Cassini formula” for Fibonacci numbers. An original Fibonacci coding/decoding method follows from the Fibonacci matrices. In contrast to classical redundant codes a basic peculiarity of the method is that it allows to correct matrix elements that can be theoretically unlimited integers. For the simplest case the correct ability of the method is equal 93.33% which exceeds essentially all well-known correcting codes.

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  • Stakhov, A.P., 2006. "Fibonacci matrices, a generalization of the “Cassini formula”, and a new coding theory," Chaos, Solitons & Fractals, Elsevier, vol. 30(1), pages 56-66.
    • Handle: RePEc:eee:chsofr:v:30:y:2006:i:1:p:56-66
    DOI: 10.1016/j.chaos.2005.12.054
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    Cited by:

    1. Emanuele Bellini & Chiara Marcolla & Nadir Murru, 2021. "An Application of p -Fibonacci Error-Correcting Codes to Cryptography," Mathematics, MDPI, vol. 9(7), pages 1-17, April.
    2. Lyes Ait-Amrane & Djilali Behloul, 2022. "Generalized hyper-Lucas numbers and applications," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(1), pages 62-75, March.
    3. Koshkin, Sergiy & Styers, Taylor, 2017. "From golden to unimodular cryptography," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 208-214.
    4. Ivana Matoušová & Pavel Trojovský, 2020. "On Coding by (2, q )-Distance Fibonacci Numbers," Mathematics, MDPI, vol. 8(11), pages 1-24, November.
    5. Kılıç, Emrah, 2009. "The generalized Pell (p,i)-numbers and their Binet formulas, combinatorial representations, sums," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 2047-2063.
    6. Basu, Manjusri & Prasad, Bandhu, 2009. "Coding theory on the m-extension of the Fibonacci p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2522-2530.
    7. Basu, Manjusri & Prasad, Bandhu, 2009. "The generalized relations among the code elements for Fibonacci coding theory," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2517-2525.
    8. Mücahit Akbiyik & Jeta Alo, 2021. "On Third-Order Bronze Fibonacci Numbers," Mathematics, MDPI, vol. 9(20), pages 1-14, October.
    9. Falcón, Sergio & Plaza, Ángel, 2007. "The k-Fibonacci sequence and the Pascal 2-triangle," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 38-49.

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