No abstract

No abstract

In this article, we generalize the well-known Gauss Pell numbers and refer to them as generalized Gauss k-Pell numbers. There are relationships discovered between the class of generalized Gauss k-Pell numbers and the typical Gauss Pell numbers. Also, we generalize the known Gauss Pell polynomials, and call such polynomials as the generalized Gauss k-Pell polynomials. We obtain relations between the class of the generalized Gauss k-Pell polynomials and the typical Gauss Pell polynomials. Furthermore, we provide matrices for the novel generalizations of these numbers and polynomials. After that, we obtain Cassini’s identities for these numbers and polynomials.

In this paper, we generalize the known Gaussian Pell-Lucas numbers, and call such numbers as the generalized Gaussian k-Pell-Lucas numbers. We obtain relations between the family of the generalized Gaussian k-Pell-Lucas numbers and the known Gaussian Pell-Lucas numbers. We generalize the known Gaussian Pell-Lucas polynomials, and call such polynomials as the generalized Gaussian k-Pell-Lucas polynomials. We obtain relations between the family of the generalized Gaussian k-Pell-Lucas polynomials and the known Gaussian Pell-Lucas polynomials. In addition, we present the new generalizations of these numbers and polynomials in matrix form. Then, we get Cassini’s identities for these numbers and polynomials.

In this paper, we introduce additive Toeplitz codes over F 4. The additive Toeplitz codes are a generalization of additive circulant codes over F 4. We find many optimal additive Toeplitz codes (OATC) over F 4. These optimal codes also contain optimal non-circulant codes, so we find new additive codes in this manner. We provide some theorems to partially classify OATC. Then, we give a new algorithm that fully classifies OATC by combining these theorems with Gaborit's algorithm. We classify OATC over F 4 of length up to 13. We obtain 2 inequivalent optimal additive toeplitz codes (IOATC) that are noncirculant codes of length 5, 92 of length 8, 2068 of length 9, and 39 of length 11. Moreover, we improve an idea related to quadratic residue codes to construct optimal and near-optimal additive Toeplitz codes over F 4 of length prime p. We obtain many optimal and near-optimal additive Toeplitz codes for some primes p from this construction.