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    m-ADIC RESIDUE CODES OVER THE RING Fq[v]/(vs - v) AND THEIR APPLICATIONS TO QUANTUM CODES
    (Rinton Press, Inc, 2022) Kuruz, Ferhat; Sari, Mustafa; Koroglu, Mehmet E.
    Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case of cyclic codes. The m-adic residue codes are the generalization of the duadic codes. The aim of this paper is to study the structure of the m-adic residue codes over the quotient ring F-q[v]/< v(s)-v >. We determine the idempotent generators of the m-adic residue codes over F-q[v]/< v(s)-v >. We obtain some parameters of optimal m-adic residue codes over F-q[v]/< v(s)-v > with respect to Griesmer bound for rings. Furthermore, we derive a condition for m-adic residue codes over F-q [v]/< v(s)-v > to contain their dual. By making use of a preserving-orthogonality Gray map, we construct a family of quantum error correcting codes from the Gray images of dual-containing m-adic residue codes over F-q[v]/< v(s)-v > and give some examples to illustrate our findings.
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    On Leonardo Pisano dual quaternions
    (Taru Publications, 2024) Dağdeviren, Ali; Kuruz, Ferhat; Catarino, Paula
    In this work, firstly we introduce the Leonardo Pisano dual quaternions combining Leonardo Pisano numbers and dual quaternions. Then we examine some fundamental properties and identities of the Leonardo Pisano dual quaternions, such as recurrence relations, generating function, summing formulas, Binet's formula, Cassini and Catalan's identities.
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    On the Horadam hybrid quaternions
    (Tbilisi Centre Math Sci, 2022) Dagdeviren, Ali; Kuruz, Ferhat
    In this study, we firstly define the Horadam hybrid quaternions and present some of their properties. Then, we define Fibonacci and Lucas hybrid quaternions, and also we study the relationship between the Fibonacci and the Lucas hybrid quaternions which connect the Fibonacci quaternions and Lucas quaternions. Furthermore, we also give some identities such as the Binet formulas and Cassini identities for Fibonacci and Lucas hybrid quaternions.
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    Pell and Pell-Lucas hybrid quaternions
    (Univ Nis, Fac Sci Math, 2023) Kuruz, Ferhat; Dagdeviren, Ali
    In this paper, we investigate Pell and Pell-Lucas numbers what new properties we can obtain by working on a new number system-hybrid quaternions. Firstly, we present some new identities of Pell and Pell-Lucas numbers and quaternions. Then, with the help of these identities and previously known identities, we get new identities on Pell and Pell-Lucas hybrid quaternions, including Binet's and Cassini's identities.
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    A study on matrices with hybrid number entries
    (Tbilisi Centre Math Sci, 2022) Erkan, Esra; Kuruz, Ferhat; Dagdeviren, Ali
    Y In this paper, we firstly introduce the matrices with hybrid numbers entries. For the sake of compatibility with the literature, we will call this set of matrices as hybrid matrices. These matrices can be regarded as a generalization of complex matrices, dual matrices, and hyperbolic matrices. We then give basic properties of hybrid matrices by writing these matrices as a combination of real matrices. Finally, we examine the real matrix representation of hybrid matrices using the base elements.