Related papers: On some batch code properties of the simplex code

A famous result of Freiman describes the structure of finite sets A of integers with small doubling property. If |A + A| <= K|A| then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K)|A|. Here…

We give a combinatorial proof of an identity that involves Eulerian numbers and was obtained algebraically by Brenti and Welker (2009). To do so, we study alcoved triangulations of dilated hypersimplices. As a byproduct, we describe the…

Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of $\mathbb{F}_q$ to construct minimal linear codes. By properties of…

This paper concerns the characterisation of second order marginals for random sets in a discrete setting. Under the instance of unit covariances, this problem possesses a combinatorial symmetry, exploited jointly in the companion paper to…

The mathematical software \texttt{GAP} (Groups, Algorithms, Programming) offers a powerful set of tools to investigate computationally group theory. Using this software package we investigate a variation of a well-known problem in…

We study the codegree isomorphism problem for finite simple groups. In particular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters. The proof is uniform for all simple groups…

Let $f(n)=\sum_k \binom nk^{-1}$. In a previous paper, we defined for a p-adic integer x that f(x) is p-definable if lim $f(x_j)$ exists in $Q_p$, where $x_j$ denotes the mod $p^j$ reduction of $x$. We proved that if p is odd, then -1 is…

In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of $\Tr(x^d)$ over $\F_{2^{m}}$ in the case where $m=2t$,…

Each number field has an associated finite abelian group, the class group, that records certain properties of arithmetic within the ring of integers of the field. The class group is well-studied, yet also still mysterious. A central…

To a function with values in the power set of a pre--ordered, separated locally convex space a family of scalarizations is given which completely characterizes the original function. A concept of a Legendre--Fenchel conjugate for set-valued…

We discuss a class of binary cyclic codes and their dual codes. The minimum distance is determined using algebraic geometry, and an application of Weil's theorem. We relate the weights appearing in the dual codes to the number of rational…

Abelian groups are classified by the existence of certain additive decompositions of group-valued functions of several variables with arity gap 2.

The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…

In this investigation of character tables of finite groups we study basic sets and associated representation theoretic data for complementary sets of conjugacy classes. For the symmetric groups we find unexpected properties of characters on…

We prove a strong general-purpose bound for the diameter of a finite group depending only on the diameters of its composition factors and the maximal exponent of a normal abelian section. There are a number of notable applications: (1) if…

Let $A$ and $B$ be subsets of a finite group $G$ and $r$ a positive integer. If for every $g\in G$, there are precisely $r$ pairs $(a,b)\in A\times B$ such that $g=ab$, then $B$ is called a code in $G$ with respect to $A$ and we write $r…

For those deformations that satisfy a certain non-degeneracy condition, we describe the structure of certain simple modules of the deformations of the subcharacter algebra of a finite group. For finite abelian groups, we prove that the…

In this paper, we define Abelian and consta-Abelian polyadic codes over rings defined as affine algebras over chain rings. For that aim, we use the classical construction via splittings and multipliers of the underlying Abelian group. We…

We address the question of constructing explicitly quasi-uniform codes from groups. We determine the size of the codebook, the alphabet and the minimum distance as a function of the corresponding group, both for abelian and some nonabelian…

We discuss conjectures related to the following two conjectures: (1) for each complex numbers x_1,...,x_n there exist rationals y_1,...,y_n \in [-2^{n-1},2^{n-1}] such that \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in…