Related papers: Designs, linear codes, plateaued functions, and th…
Boolean networks are popular tools for the exploration of qualitative dynamical properties of biological systems. Several dynamical interpretations have been proposed based on the same logical structure that captures the interactions…
Interaction graphs provide an important qualitative modeling approach for System Biology. This paper presents a novel approach for construction of interaction graph with the help of Boolean function decomposition. Each decomposition part…
Generally, combinatorial design concerns with the arrangement of a finite set of elements into patterns (subsets, words, arrays) according to specified rules. The usefulness of this design method is that the number of input combination can…
A fascinating topic of combinatorics is $t$-designs, which have a very long history. The incidence matrix of a $t$-design generates a linear code over GF$(q)$ for any prime power $q$, which is called the linear code of the $t$-design over…
Boolean functions can be represented in many ways including logical forms, truth tables, and polynomials. Additionally, Boolean functions have different canonical representations such as minimal disjunctive normal forms. Other canonical…
We introduce combinatorial interpretability, a methodology for understanding neural computation by analyzing the combinatorial structures in the sign-based categorization of a network's weights and biases. We demonstrate its power through…
In this paper we study the relationship between vectorial (Boolean) functions and cyclic codes in the context of algebraic attacks. We first derive a direct link between the annihilators of a vectorial function (in univariate form) and…
We investigate point-line geometries whose singular subspaces correspond to binary equidistant codes. The main result is a description of automorphisms of these geometries. In some important cases, automorphisms induced by non-monomial…
This work provides a complete characterization of the solutions of a linear interpolation problem for vector polynomials. The interpolation problem consists in finding n scalar polynomials such that an equation involving a linear…
Large sets of combinatorial designs has always been a fascinating topic in design theory. These designs form a partition of the whole space into combinatorial designs with the same parameters. In particular, a large set of block designs,…
We regard explanations as a blending of the input sample and the model's output and offer a few definitions that capture various desired properties of the function that generates these explanations. We study the links between these…
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves…
This paper considers the construction of optimal designs due to Hoel and Levine and Guest. It focuses on the relation between the theory of the uniform approximation of functions and the optimality of the designs. Some application to…
We discuss general models of resource-sharing computations, with emphasis on the combinatorial structures and concepts that underlie the various deadlock models that have been proposed, the design of algorithms and deadlock-handling…
Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by $\mathcal{C}(f)=\left\{ {\rm Tr}(af(x)+bx)_{x \in \mathbb{F}_{q^m}^*}: a,b \in…
We determine all composition-closed equational classes of Boolean functions. These classes provide a natural generalization of clones and iterative algebras: they are closed under composition, permutation and identification…
The aim of this study is to construct and compose an instructional design in combinatorial learning, particularly in the concept of counting. A composed design is expected to optimize students' combinatorial-thinking skill. This research…
This paper defends an augmented cognitively oriented "generic-design hypothesis": There are both significant similarities between the design activities implemented in different situations and crucial differences between these and other…
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
A novel polynomial expansion method of symmetric Boolean functions is described. The method is efficient for symmetric Boolean function with small set of valued numbers and has the linear complexity for elementary symmetric Boolean…