Reversible codes and applications to DNA codes over F42t[u]/(u2−1) F_{4^{2t}}[u]/(u^2-1)

Let $n \geq 1$ be a fixed integer. Within this study, we present a novel approach for discovering reversible codes over rings, leveraging the concept of $r$-glifted polynomials. This technique allows us to achieve optimal reversible codes. As we extend our methodology to the domain of DNA codes, we establish a correspondence between $4t$-bases of DNA and elements within the ring $R_{2t} = F_{4^{2t}}[u]/(u^{2}-1)$. By employing a variant of $r$-glifted polynomials, we successfully address the challenges of reversibility and complementarity in DNA codes over this specific ring. Moreover, we are able to generate reversible and reversible-complement DNA codes that transcend the limitations of being linear cyclic codes generated by a factor of $x^n-1$

A New Algebraic Structure of Complex Pythagorean Fuzzy Subfield

The concept of complex Pythagorean fuzzy set $(\mathbf {CPFS})$ is recent development in the field of fuzzy set $(\mathbf {FS})$ theory. The significance of this concept lies in the fact that this theory assigned membership grades $\psi$ and non-membership grades $\hat {\psi }$ from unit circle in plane, i.e., in the form of a complex number with the condition $(\psi)^{2}+ (\hat {\psi })^{2}\le 1$ instead from [0, 1] interval. This is an expressive technique for dealing with uncertain circumstances. The aim of this study is to proceed the classification of the unique framework of $\mathbf {CPFS}$ in algebraic structure that is field theory and examine its numerous algebraic features. Also, we initiate the important examples and results of certain field. Furthermore, we illustrate that every complex Pythagorean fuzzy subfield ( $\mathbf {CPFSF}$ ) generates two Pythagorean fuzzy subfields $(\mathbf {PFSFs})$ . We also prove many useful algebraic aspects of this notion for a $\mathbf {CPFSF}$ . Moreover, we demonstrate that intersection of two complex Pythagorean fuzzy subfields $(\mathbf {CPFSFs})$ is also $\mathbf {CPFSF}$ . Additionally, we discuss the novel idea of level subsets of $\mathbf {CPFSFs}$ and demonstrate that level subset of $\mathbf {CPFSF}$ form subfield. Additionally, we improve the application of this theory to show the concept of the direct product of two $\mathbf {CPFSFs}$ is also a $\mathbf {CPFSF}$ and produce several novel results on direct product of $\mathbf {CPFSFs}$ . Finally, we explore the homomorphic images and inverse images of $\mathbf {CPFSFs}$

Characterization of Lie-Type Higher Derivations of von Neumann Algebras with Local Actions

Let m and n be fixed positive integers. Suppose that A is a von Neumann algebra with no central summands of type I1, and Lm:A→A is a Lie-type higher derivation. In continuation of the rigorous and versatile framework for investigating the structure and properties of operators on Hilbert spaces, more facts are needed to characterize Lie-type higher derivations of von Neumann algebras with local actions. In the present paper, our main aim is to characterize Lie-type higher derivations on von Neumann algebras and prove that in cases of zero products, there exists an additive higher derivation ϕm:A→A and an additive higher map ζm:A→Z(A), which annihilates every (n−1)th commutator pn(S1,S2,⋯,Sn) with S1S2=0 such that Lm(S)=ϕm(S)+ζm(S)forallS∈A. We also demonstrate that the result holds true for the case of the projection product. Further, we discuss some more related results