scholarly journals On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

10.37236/969 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Jörn Quistorff
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Finite Sets ◽  
Open Problems ◽  
Latin Rectangle ◽  
Mixed Codes ◽  
Special Case

Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

scholarly journals Lower Bounds for the Football Pool Problem for 7 and 8 Matches

10.37236/945 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas
Keyword(s):  
Lower Bounds ◽  
Covering Radius ◽  
Linear Inequalities ◽  
Minimal Cardinality ◽  
Covering Code ◽  
Ternary Code ◽  
Systems Of Linear Inequalities

Let $k_3(n)$ denote the minimal cardinality of a ternary code of length $n$ and covering radius one. In this paper we show $k_3(7)\ge 156$ and $k_3(8)\ge 402$ improving on the best previously known bounds $k_3(7)\ge 153$ and $k_3(8)\ge 398$. The proofs are founded on a recent technique of the author for dealing with systems of linear inequalities satisfied by the number of elements of a covering code, that lie in $k$-dimensional subspaces of F${}_3^n$.

scholarly journals Lower Bounds for $q$-ary Codes with Large Covering Radius

10.37236/222 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Immanuel Halupczok ◽  
Jan-Christoph Schlage-Puchta
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Efficient Method ◽  
Winning Strategy ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Original Proof ◽  
Sphere Covering

Let $K_q(n,R)$ denote the minimal cardinality of a $q$-ary code of length $n$ and covering radius $R$. Recently the authors gave a new proof of a classical lower bound of Rodemich on $K_q(n,n-2)$ by the use of partition matrices and their transversals. In this paper we show that, in contrast to Rodemich's original proof, the method generalizes to lower-bound $K_q(n,n-k)$ for any $k>2$. The approach is best-understood in terms of a game where a winning strategy for one of the players implies the non-existence of a code. This proves to be by far the most efficient method presently known to lower-bound $K_q(n,R)$ for large $R$ (i.e. small $k$). One instance: the trivial sphere-covering bound $K_{12}(7,3)\geq 729$, the previously best bound $K_{12}(7,3)\geq 732$ and the new bound $K_{12}(7,3)\geq 878$.

scholarly journals On mixed codes with covering radius 1 and minimum distance 2

2007 ◽  
Vol 29 ◽  
pp. 11-15
Author(s):  
W. Haas ◽  
J. Quistorff
Keyword(s):  
Minimum Distance ◽  
Covering Radius ◽  
Mixed Codes

The application of linear complexity of sequences to lower bounds on the minimum distance of cyclic codes

1990 ◽  
Vol 7 (4) ◽  
pp. 312-316
Author(s):  
Li Yuanxing ◽  
Liang Chuanjia
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Linear Complexity ◽  
Cyclic Codes

Mutual Embeddings

2015 ◽  
Vol 15 (01n02) ◽  
pp. 1550001
Author(s):  
ILKER NADI BOZKURT ◽  
HAI HUANG ◽  
BRUCE MAGGS ◽  
ANDRÉA RICHA ◽  
MAVERICK WOO
Keyword(s):  
Iterative Methods ◽  
Lower Bounds ◽  
Linear Equations ◽  
Graph Embedding ◽  
Open Problems ◽  
Linear Arrays ◽  
Systems Of Linear Equations

This paper introduces a type of graph embedding called a mutual embedding. A mutual embedding between two n-node graphs [Formula: see text] and [Formula: see text] is an identification of the vertices of V1 and V2, i.e., a bijection [Formula: see text], together with an embedding of G1 into G2 and an embedding of G2 into G1 where in the embedding of G1 into G2, each node u of G1 is mapped to π(u) in G2 and in the embedding of G2 into G1 each node v of G2 is mapped to [Formula: see text] in G1. The identification of vertices in G1 and G2 constrains the two embeddings so that it is not always possible for both to exhibit small congestion and dilation, even if there are traditional one-way embeddings in both directions with small congestion and dilation. Mutual embeddings arise in the context of finding preconditioners for accelerating the convergence of iterative methods for solving systems of linear equations. We present mutual embeddings between several types of graphs such as linear arrays, cycles, trees, and meshes, prove lower bounds on mutual embeddings between several classes of graphs, and present some open problems related to optimal mutual embeddings.

scholarly journals Bounding the Size and Probability of Epidemics on Networks

2008 ◽  
Vol 45 (2) ◽  
pp. 498-512 ◽  
Author(s):  
Joel C. Miller
Keyword(s):  
Infectious Disease ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Marginal Probability ◽  
Transmission Properties ◽  
Disease Spreading ◽  
Special Case

We consider an infectious disease spreading along the edges of a network which may have significant clustering. The individuals in the population have heterogeneous infectiousness and/or susceptibility. We define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility. For a given distribution of out-transmissibility, we find the conditions which give the upper (or lower) bounds on the size and probability of an epidemic, under weak assumptions on the transmission properties, but very general assumptions on the network. We find similar bounds for a given distribution of in-transmissibility (the marginal probability of being infected by a neighbor). We also find conditions giving global upper bounds on the size and probability. The distributions leading to these bounds are network independent. In the special case of networks with high girth (locally tree-like), we are able to prove stronger results. In general, the probability and size of epidemics are maximal when the population is homogeneous and minimal when the variance of in- or out-transmissibility is maximal.

New Results on Codes with Covering Radius 1 and Minimum Distance 2

2005 ◽  
Vol 35 (2) ◽  
pp. 241-250 ◽  
Author(s):  
Patric R. J. �sterg�rd ◽  
J�rn Quistorff ◽  
Alfred Wassermann
Keyword(s):  
Minimum Distance ◽  
Covering Radius

scholarly journals On the history of van der Waerden's theorem on arithmetic progressions

2001 ◽  
Vol 32 (4) ◽  
pp. 335-341
Author(s):  
Tom C. Brown ◽  
Jau-Shyong Peter Shiue
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progressions ◽  
Upper And Lower Bounds ◽  
Open Problems ◽  
History Of

In this expository note, we discuss the celebrated theorem known as ``van der Waerden's theorem on arithmetic progressions", the history of work on upper and lower bounds for the function associated with this theorem, a number of generalizations, and some open problems.

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

Lower bounds for cutting planes proofs with small coefficients

1997 ◽  
Vol 62 (3) ◽  
pp. 708-728 ◽  
Author(s):  
Maria Bonet ◽  
Toniann Pitassi ◽  
Ran Raz
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Proof System ◽  
Small Weight ◽  
Deduction Rule ◽  
Special Case ◽  
Exponential Lower Bound

AbstractWe consider small-weight Cutting Planes (CP*) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution.We also prove the following two theorems: (1) Tree-like CP* proofs cannot polynomially simulate non-tree-like CP* proofs. (2) Tree-like CP* proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other.Our proofs also work for some generalizations of the CP* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

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