In the article, we use the subset sum formula over a finite abelian group on the product of finite groups to derive the number of restricted partitions of elements in the group and to count the number of compositions over finite abelian groups. Later, we apply the formula for the multisubset sum problem on a group $\mathbb{Z}_n$ to produce a new technique for studying restricted partitions of positive integers. 2020 Mathematics Subject Classification. 05A17, 11P81

We present an application of generalized strong complete mappings to construction of a family of mutually orthogonal Latin squares. We also determine a cycle structure of such mapping which form a complete family of MOLS. Many constructions of generalized strong complete mappings over an extension of finite field are provided.

We show the existence of many infinite classes of permutations over finite fields and bent functions by extending the notion of linear translators, introduced by Kyureghyan (J. Combin. Theory Ser. A 118(3), 1052–1061, 2011). We call these translators Frobenius translators since the derivatives of f:Fpn→Fpk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f:{\mathbb F}_{p^{n}} \rightarrow {\mathbb F}_{p^{k}}$\end{document}, where n = rk, are of the form f(x+uγ)−f(x)=upib\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x+u\gamma )-f(x)=u^{p^{i}}b$\end{document}, for a fixed b∈Fpk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b \in {\mathbb F}_{p^{k}}$\end{document} and all u∈Fpk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u \in {\mathbb F}_{p^{k}}$\end{document}, rather than considering the standard case corresponding to i = 0. It turns out that Frobenius translators correspond to standard linear translators of an exponentiated version of f, namely to fpk−i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f^{p^{k-i}}$\end{document} with respect to bpk−i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b^{p^{k-i}}$\end{document}. Nevertheless, this concept turns out to be useful for providing further explicit specification of a rather rare family {f} of quadratic polynomials (especially sparse ones) admitting linear translators. In this direction, we solve a few open problems in the recent article (Cepak et al., Finite Fields Appl. 45, 19–42, 2017) concerning the existence and an exact specification of f admitting classical linear translators. In addition, an open problem introduced in Hodžić et al. (2018), of finding a triple of bent functions f1,f2,f3 such that their sum f4 is bent and that the sum of their duals satisfies f1∗+f2∗+f3∗+f4∗=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{1}^{*}+f_{2}^{*}+f_{3}^{*}+f_{4}^{*}=1$\end{document}, is also resolved. We also specify two huge families of permutations over Fpn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb F}_{p^{n}}$\end{document} related to the condition that G(y)=−L(y)+(y+δ)s−(y+δ)pks\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G(y)=-L(y)+(y+\delta )^{s}-(y+\delta )^{p^{k}s}$\end{document} permutes the set S={β∈Fpn:Trkn(β)=0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal S}=\{\beta \in {\mathbb F}_{p^{n}}: T{r_{k}^{n}}(\beta )=0\}$\end{document}, where n = 2k and p > 2. Finally, we give some generalizations of constructions of bent functions in Mesnager et al. (2017) and describe some new bent families using the permutations found in Cepak et al. (Finite Fields Appl. 45, 19–42, 2017).