scholarly journals Ternary Covering Codes

10.37236/1281 ◽  
1996 ◽  
Vol 3 (2) ◽  
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.

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 Covering Codes for Hats-on-a-line

10.37236/1047 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Sarang Aravamuthan ◽  
Sachin Lodha
Keyword(s):  
Exact Solution ◽  
Optimal Strategy ◽  
Arbitrary Number ◽  
Optimization Problems ◽  
Covering Radius ◽  
Binary Codes ◽  
Covering Codes ◽  
Natural Extensions ◽  
Simulated Game

We consider a popular game puzzle, called Hats-on-a-line, wherein a warden has $n$ prisoners, each one wearing a randomly assigned black or white hat, stand in a line. Thus each prisoner can see the colors of all hats before him, but not his or of those behind him. Everyone can hear the answer called out by each prisoner. Based on this information and without any further communication, each prisoner has to call out his hat color starting from the back of the line. If he gets it right, he is released from the prison, otherwise he remains incarcerated forever. The goal of the team is to devise a strategy that maximizes the number of correct answers. A variation of this problem asks for the solution for an arbitrary number of colors. In this paper, we study the standard Hats-on-a-line problem and its natural extensions. We demonstrate an optimal strategy when the seeing radius and/or the hearing radius are limited. We show for certain orderings that arise from a (simulated) game between the warden and prisoners, how this problem relates to the theory of covering codes. Our investigations lead to two optimization problems related to covering codes in which one leads to an exact solution (for binary codes). For instance, we show that for $0 < k < n$, $(n-k-d) \le \alpha_m n$ where $d = t(n-k, m^k, m)$ is the minimum covering radius of an $m$-ary code of length ($n-k$) and size $m^k$ and $$\alpha_m = {\log m\over \log (m^2 -m +1)}.$$

scholarly journals New Bounds on the Minimum Average Distance of Binary Codes

2010 ◽  
Vol 2 (3) ◽  
pp. 489
Author(s):  
M. Basu ◽  
S. Bagchi
Keyword(s):  
Lower Bounds ◽  
Binary Code ◽  
Hamming Distance ◽  
Average Distance ◽  
Upper And Lower Bounds ◽  
Binary Codes ◽  
Large N

The minimum average Hamming distance of binary codes of length n and cardinality M is denoted by b(n,M). All the known lower bounds b(n,M) are useful when M is at least of size about 2n-1/n . In this paper, for large n, we improve upper and lower bounds for b(n,M). Keywords: Binary code; Hamming distance; Minimum average Hamming distance. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i3.2708                  J. Sci. Res. 2 (3), 489-493 (2010) 

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.

Bounds for abnormal binary codes with covering radius one

1991 ◽  
Vol 37 (2) ◽  
pp. 372-375 ◽  
Author(s):  
H.S. Honkala ◽  
H.O. Hamalainen
Keyword(s):  
Covering Radius ◽  
Binary Codes
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