In this note, we show that the method of Croot, Lev, and Pach can be used to bound size a subset F n q  Fqn with no three terms in arithmetic progression by c n  cn c<q c<q . For q=3 q=3 , problem finding largest F n 3  F3n is called cap set problem. Previously best known upper for affine problem, due Bateman Katz, was on order n −1−ϵ 3 n  n−1−ϵ3n

Abstract We prove new upper bounds on the smallest size of affine blocking sets, that is, sets points in a finite space intersect every subspace fixed codimension. show an equivalence between with respect to codimension‐2 subspaces are generated by taking union lines through origin, and strong corresponding projective space, which turn equivalent minimal codes. Using this equivalence, we improve current best set spaces over fields at least 3. Furthermore, using coding theoretic techniques,...

Noisy hardware forms one of the main hurdles to realization a near-term quantum internet. Distillation protocols allows overcome this noise at cost an increased overhead. We consider here experimentally relevant class distillation protocols, which distill <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> xmlns:xlink="http://www.w3.org/1999/xlink">k</i> end-to-end entangled pairs using bilocal Clifford operations, single round...

This PhD thesis is concerned with SDP bounds for codes: upper (non)-binary error correcting codes and lower covering codes. The methods are based on the method of Schrijver that uses triple distances in stead pairs as classical Delsarte bound. main topics discussed are: 1) Block-diagonalisation matrix *-algebras, 2) Terwilliger-algebra nonbinary Hamming scheme (including an explicit block-diagonalisation), 3) SDP-bounds (nonbinary) error-correcting computational results), 4) Discussion...

Let A(n, d) be the maximum number of 0, 1 words length n, any two having Hamming distance at least d.We prove A(20, 8) = 256, which implies that quadruply shortened Golay code is optimal.Moreover, we show A( 18,6) ≤ 673,A(19,6) 1237,A(20,6) 2279,A(23,6) 13674,A(19,8) 135,A(25,8) 5421,A(26,8) 9275,A(21,10) 47,A(22,10) 84, A(24, 10) 268, A(25, 466, A(26, 836, A(27, 1585, 12) 55, and 96.The method based on positive semidefiniteness matrices derived from quadruples words.This can put as...

Entanglement distillation is an essential building block in quantum communication protocols. Here, we study the class of near-term implementable protocols that use bilocal Clifford operations followed by a single round communication. We introduce tools to enumerate and optimise over all for up <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>5</mml:mn></mml:math> (not necessarily equal) Bell-diagonal states using commodity desktop computer....

Given a graph G = (V,E) and "cost function" (provided by an oracle), the problem [PCliqW] consists in finding partition into cliques of V(G) minimum cost. Here, cost is sum costs partition. We provide polynomial time dynamic program for case where interval f belongs to subclass submodular set functions, which we call "value-polymatroidal". This provides common solution various generalizations coloring co-interval graphs such as max-coloring, "Greene-Kleitman's dual", probabilist chromatic...

There are several notions of gonality for graphs. The divisorial dgon(G) a graph G is the smallest degree divisor positive rank in sense Baker–Norine. stable sgon(G) minimum finite harmonic morphism from refinement to tree, as defined by Cornelissen, Kato and Kool. We show that computing NP-hard reduction maximum independent set problem vertex cover problem, respectively. Both constructions moreover APX-hard.

We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. Graphs for which equality holds include grid graphs and complete multipartite graphs. treewidth bound also metric (tropical curves) constructing any positive rank divisor on same degree subdivision underlying combinatorial graph. Finally, we show related notion defined Caporaso stable as introduced Cornelissen et al.

Noisy hardware forms one of the main hurdles to realization a near-term quantum internet. Distillation protocols allows overcome this noise at cost an increased overhead. We consider here experimentally relevant class distillation protocols, which distill $n$ $k$ end-to-end entangled pairs using bilocal Clifford operations, single round communication and possible final local operation depending on observed measurement outcomes. In case permutationally invariant depolarizing input states, we...

The Shannon capacity of a graph G is the maximum asymptotic rate at which messages can be sent with zero probability error through noisy channel confusability . This extensively studied parameter disregards fact that on atomic scales, nature behaves in line quantum mechanics. Entanglement, arguably most counterintuitive feature theory, turns out to useful resource for communication across channels. Recently [Leung D, Mančinska L, Matthews W, Ozols M, Roy A (2012) Commun Math Phys...

We call a matrix A nearly totally unimodular if it can be obtained from $\tilde{A}$ by adding to each row of an integer multiple some fixed $a^{\transp}$ $\tilde{A}$. For vector b and A, we denote $P_{A,b}$ the hull set $\{x\in\mathbb{R}^n\mid Ax\leq b\}$. show that has decomposition property find given $x\in kP_{A,b}$ in polynomial time. An interesting special case plays role many cyclic scheduling problems is when circular-ones matrix. In this case, nonnegative k x, testing finding x into...

For a matrix *-algebra B, consider the A consisting of symmetric tensors in n-fold tensor product B. Examples such algebras coding theory include Bose-Mesner algebra and Terwilliger (non)binary Hamming cube, arising SDP-hierarchies for bounds using moment matrices. We give computationally efficient block diagonalization terms given work out some examples, including binary- nonbinary cube. As tool we use basic facts about representations group.

We study the capacitated $k$-facility location problem, in which we are given a set of clients with demands, facilities capacities and constant number $k$. It costs $f_i$ to open facility $i$, $c_{ij}$ for $i$ serve one unit demand from client $j$. The objective is at most $k$ serving all demands satisfying capacity constraints while minimizing sum service opening costs. In this paper, give first fully polynomial time approximation scheme (FPTAS) single-sink (single-client) problem. Then,...

An integer packing set is a of non-negative vectors with the property that, if vector $x$ in set, then every $y$ $y \leq x$ as well. Integer sets appear naturally Optimization. In fact, points any polyhedron an set. The main result this paper that sets, ordered by inclusion, form well-quasi-ordering. This allows us to answer question recently posed Bodur et al. we prove k-aggregation closure again polyhedron. generality our provide generalization non-polyhedral sets: downset $\mathbb{R}^n_+$

We show that any subset of $\mathbb{Z}_p^n$ ($p$ an odd prime) without $3$-term arithmetic progression has size $O(p^{cn})$, where $c:=1-\frac{1}{18\log p}&lt;1$. In particular, we find upper bound $O(2.84^n)$ on the maximum affine cap in $GF(3)^n$.