Vadiraja Bhatta G. R | Searcharxiv

arXiv2025

Reversible and Reversible-Complement Double Cyclic Codes over F4+vF4 and its Application to DNA Codes

In this article, we study the algebraic structure of double cyclic codes of length $(m, n)$ over $\mathbb{F}_4$ and we give a necessary and sufficient condition for a double cyclic code over $\mathbb{F}_4$ to be reversible. Also, we determine the algebraic structure of double cyclic codes of length $(m, n)$ over $\mathbb{F}_4+v\mathbb{F}_4$ with $v^2=v$, satisfying the reverse constraint and the reverse-complement constraint. Then we establish a one-to-one correspondence $ψ$ between the 16 DNA double pairs $S_{D_{16}} $ and the 16 elements of the finite ring $\mathbb{F}_4+v\mathbb{F}_4$. We also discuss the GC-content of DNA double cyclic codes.

math.RA↗

arXiv2025

Some more constructions of $n-$cycle permutation polynomials

$ n-$cycle permutation polynomials with small n have the advantage that their compositional inverses are efficient in terms of implementation. These permutation polynomials have significant applications in cryptography and coding theory. In this article, we propose criteria for the construction of $ n-$cycle permutation using linearized polynomial $ L(x) $ for larger $ n $. Furthermore, we investigate and generalize certain novel forms of $ n-$cycle permutation polynomials. Finally, we demonstrate our approach by constructing explicit $ n-$cycle permutation of the form $ L(x)+γh(Tr_{q^{m}/q}(x)) $, and $ G(x)+γf(x) $ with a Boolean function $ f(x) $. The polynomial $ x^{d}+γf(x) $ with $ f(x) $ being a Boolean function is shown to be quadruple and quintuple permutation polynomials. Moreover, linear binomial triple-cycle permutation polynomials are constructed.

math.RA↗