Related papers: On the trifference problem for linear codes

The classic trisecant lemma states that if $X$ is an integral curve of $\PP^3$ then the variety of trisecants has dimension one, unless the curve is planar and has degree at least 3, in which case the variety of trisecants has dimension 2.…

We show that $n$ is almost perfect if and only if $I(n) - 1 < D(n) \leq I(n)$, where $I(n)$ is the abundancy index of $n$ and $D(n)$ is the deficiency of $n$. This criterion is then extended to the case of integers $m$ satisfying $D(m)>1$.

A linear graph code is a family $\mathcal{C}$ of graphs on $n$ vertices with the property that the symmetric difference of the edge sets of any two graphs in $\mathcal{C}$ is also the edge set of a graph in $\mathcal{C}$. In this article,…

Convolutional codes with a maximum distance profile attain the largest possible column distances for the maximum number of time instants and thus have outstanding error-correcting capability especially for streaming applications. Explicit…

Let $n$ be a sufficiently large integer with $n\equiv 0\pmod 4$ and let $F_i \subseteq{[n]\choose 4}$ where $i\in [n/4]$. We show that if each vertex of $F_i$ is contained in more than ${n-1\choose 3}-{3n/4\choose 3}$ edges, then $\{F_1,…

Any $n$-vertex $3$-graph with minimum codegree at least $\lfloor n/3\rfloor$ must have a spanning tight component, but immediately below this threshold it is possible for no tight component to span more than $\lceil 2n/3\rceil$ vertices.…

The celebrated Andr\'{a}sfai--Erd\H{o}s--S\'{o}s Theorem from 1974 shows that every $n$-vertex triangle-free graph with minimum degree greater than $2n/5$ must be bipartite. Its extensions to $3$-uniform hypergraphs without the generalized…

In this paper we confirm a conjecture of F\"uredi, Jiang, and Seiver, and determine an exact formula for the Tur\'an number $ex_3(n; P_3^3)$ of the 3-uniform linear path $P^3_3$ of length 3, valid for all $n$. It coincides with the…

Let $\mathbb{F}$ be a field. Denote by $t_n(\mathbb{F})$ the greatest possible dimension for a vector space of $n$-by-$n$ matrices over $\mathbb{F}$ in which every element is triangularizable over $\mathbb{F}$. It was recently proved that…

We prove that a smooth complex projective threefold with a K\"ahler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef…

Given a graph $H$, a perfect $H$-factor in a graph $G$ is a collection of vertex-disjoint copies of $H$ spanning $G$. K\"uhn and Osthus showed that the minimum degree threshold for a graph $G$ to contain a perfect $H$-factor is either given…

There is a constant c > 0 such that for every $\epsilon \in (0,1)$ and $n \geq 1/\epsilon^2$, the following holds. Any mapping from the $n$-point star metric into $\ell_1^d$ with bi-Lipschitz distortion $1+\epsilon$ requires dimension $$d…

In this paper implicit and explicit exact difference schemes (EDS) for system $\textbf{x}' = A\textbf{x}$ of three linear differential equations with constant coefficients are constructed. Numerical simulations for stiff problem and for…

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…

The Golomb-Welch conjecture (1968) states that there are no $e$-perfect Lee codes in $\mathbb{Z}^n$ for $n\geq 3$ and $e\geq 2$. This conjecture remains open even for linear codes. A recent result of Zhang and Ge establishes the…

Given positive integers k\geq 3 and r where k/2 \leq r \leq k-1, we give a minimum r-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an…

A linear system of plane curves satisfying multiplicity conditions at points in general position is called special if the dimension is larger than the expected dimension. A (-1) curve is an irreducible curve with self intersection -1 and…

Let $F = (U,E)$ be a graph and $\mathcal{H} = (V,\mathcal{E})$ be a hypergraph. We say that $\mathcal{H}$ contains a Berge-$F$ if there exist injections $\psi:U\to V$ and $\varphi:E\to \mathcal{E}$ such that for every $e=\{u,v\}\in E$,…

We construct constant-sized ensembles of linear error-correcting codes over any fixed alphabet that can correct a given fraction of adversarial erasures at rates approaching the Singleton bound arbitrarily closely. We provide several…

Denote by $F_5$ the $3$-uniform hypergraph on vertex set $\{1,2,3,4,5\}$ with hyperedges $\{123,124,345\}$. Balogh, Butterfield, Hu, and Lenz proved that if $p > K \log n / n$ for some large constant $K$, then every maximum $F_5$-free…