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

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${}_3^n$.

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

The Electronic Journal of Combinatorics ◽

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.

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

The Electronic Journal of Combinatorics ◽

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 $K_{12}(7,3)\geq 878$.

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Ternary Covering Codes

The Electronic Journal of Combinatorics ◽

10.37236/1281 ◽

1996 ◽

Vol 3 (2) ◽

Cited By ~ 1

Author(s):

Laurent Habsieger

Keyword(s):

Lower Bounds ◽

Covering Radius ◽

Linear Inequalities ◽

Characteristic Functions ◽

Binary Codes ◽

Covering Codes ◽

Congruence Properties

In [5], we studied binary codes with covering radius one via their characteristic functions. This gave us an easy way of obtaining congruence properties and of deriving interesting linear inequalities. In this paper we extend this approach to ternary covering codes. We improve on lower bounds for ternary $1$-covering codes, the so-called football pool problem, when $3$ does not divide $n-1$. We also give new lower bounds for some covering codes with a covering radius greater than one.

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Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

Advances in Applied Probability ◽

10.1017/apr.2020.40 ◽

2020 ◽

Vol 52 (4) ◽

pp. 1249-1283

Author(s):

Masatoshi Kimura ◽

Tetsuya Takine

Keyword(s):

Markov Chains ◽

Stationary Distribution ◽

Continuous Time ◽

State Transition ◽

Convex Polytope ◽

Linear Inequalities ◽

Systems Of Linear Inequalities ◽

Free Sets ◽

Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

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