Related papers: The nonexistence of an additive quaternary [15,5,9…

The Gelfand-Kirillov dimension measures the asymptotic growth rate of algebras. For every associative dialgebra $\mathcal{D}$, the quotient $\mathcal{A}_\mathcal{D}:=\mathcal{D}/\mathsf{Id}(S)$, where $\mathsf{Id}(S)$ is the ideal of…

We prove that there are no Wieferich's primes $q=2p+1$ where $p \equiv 3 \pmod{4}$ is a prime number

In a previous work, we introduced the notion of uniform generators of the Brauer group and proved that general diagonal cubic surfaces do not have such generators. In this paper, we prove that a similar non-existence result holds for affine…

In this paper, for an even integer $n\geq 4$ and any positive integer $k$ with ${\rm gcd}(n/2,k)={\rm gcd}(n/2-k,2k)=d$ being odd, a class of $p$-ary codes $\mathcal{C}^k$ is defined and their weight distribution is completely determined,…

Answering a question of Ma, Siegert, and Dydak we show that there is no universal proper metric space for the asymptotic dimension $n\ge1$.

Linear programming bounds provide an elegant method to prove optimality and uniqueness of an (n,N,t) spherical code. However, this method does not apply to the parameters (4,10,1/6). We use semidefinite programming bounds instead to show…

Let $p\equiv1\pmod3$ be a prime . We study several topics on additive decompositions concerning the set $C_p$ of all non-zero cubes in the finite field of $p$ elements. For example, we show that when $p>184291$ , the set $C_p$ has no…

We consider the even monic degree-$10$ second cuboid polynomial $Q_{p,q}(t)\in\mathbb{Z}[t]$ depending on coprime integers $p\neq q>0$. We exclude the existence of a splitting of type $5+5$ over $\mathbb{Q}$, i.e., a factorization of…

For every even positive integer $k\ge 4$ let $f(n,k)$ denote the minimim number of colors required to color the edges of the $n$-dimensional cube $Q_n$, so that the edges of every copy of $k$-cycle $C_k$ receive $k$ distinct colors.…

In connection with optimal state determination for two qubits, the question was raised about the maximum number of pairwise complementary reductions. The main result of the paper tells that the maximum number is 4, that is, if A1, A2,... Ak…

This paper is devoted to the complete algebraic and geometric classification of complex 4 and 5-dimensional antiassociative algebras. In particular, we proved that the variety of complex 4-dimensional antiassociative algebras has dimension…

We show that in a tracial and finitely generated $W^\ast$-probability space existence of conjugate variables excludes algebraic relations for the generators. Moreover, under the assumption of maximal non-microstates free entropy dimension,…

In this paper, we study the existence of $(v,m,k,\lambda)$-strong external difference families (SEDFs). We use character-theoretic techniques to show that no SEDF exists when $v$ is prime, $k > 1$ and $m > 2$. In the case where $v$ is the…

The punctured dodecacode is an additive $4$-ary code of length $11$ and distance $5$ which is uniformly packed. We show that a code with the same weight distribution is equivalent to it. This code is also shown to be nonlinear. We also…

We consider Diophantine quintuples $\{a, b, c, d, e\}$, sets of distinct positive integers the product of any two elements of which is one less than a perfect square. Triples of the first kind are the subsets $\{a, b, d\}$ with $d> b^{5}$.…

A 3-$(n,4,1)$ packing design consists of an $n$-element set $X$ and a collection of $4$-element subsets of $X$, called {\it blocks}, such that every $3$-element subset of $X$ is contained in at most one block. The packing number of…

Let ${\cal C}$ be a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code of length $n > 3$. We prove that if the binary Gray image of ${\cal C}$, $C=\Phi({\cal C})$, is a 1-perfect nonlinear code, then ${\cal C}$ cannot be a…

Does $14$ have a friend? Until now, this has been an open question. In this note, we prove that a potential friend $F$ of $14$ is an odd, non-square positive integer. $7$ appears in the prime factorization of $F$ with an even exponent while…

This paper investigates the existence and properties of spherical $5$-designs of minimal type. We focus on two cases: tight spherical $5$-designs and antipodal spherical $4$-distance $5$-designs. We prove that a tight spherical $5$-design…

We investigate the maximum number \( L_{\mathrm{rk}}(n, m, k, q) \) of distinct nonzero rank weights that an \( \mathbb{F}_{q^m} \)-linear rank-metric code of dimension \( k \) in \( \mathbb{F}_{q^m}^n \) can attain. We determine the exact…