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.

Abstract Given are necessary conditions for a permutation polynomial to be the derivative of a planar mapping. These conditions are not sufficient and there might exist permutation polynomials which are not derivatives of some planar mapping satisfying these conditions. For the first time we show that there is a close connection between two seemingly unrelated structures, namely planar and complete mappings. It is shown that any planar mapping induces a sequence of complete mappings having some additional interesting properties. Furthermore, a class of almost planar mappings over extension fields is introduced having the property that its derivatives are permutations in most of the cases. This class of functions then induces many infinite classes of complete mappings (permutations) as well.

Abstract Automatic creation of school timetables is a complex problem when it involves defining specific constraints and requirements. This paper presents a domain-specific language called REDOSPLAT which supports such definitions in a readable format. REDOSPLAT can be used when timetable programs with a graphical user interface are limited or too cluttered to express specific domain features. It uses customised, readable notation instead of technically oriented data formats which are usually used in timetable problems. The desired timetabling requirements are defined using sentences whose syntax is close to the syntax of spoken language. This paper encompasses the entire language syntax and semantics, the way a sentence is transformed into data structures which describe timetabling requirements, and the way the obtained data structures are transformed into models needed for different problem-solving algorithms. For the latter, REDOSPLAT is using integer linear programming to resolve the timetabling problem.

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).