Lower Bounds for Binary Codes of Covering Radius One

2007 ◽  
Vol 53 (8) ◽  
pp. 2880-2881 ◽  
Author(s):  
W. Haas
Keyword(s):  
Lower Bounds ◽  
Covering Radius ◽  
Binary Codes

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 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.

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${}_3^n$.

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

scholarly journals Lower bounds for q-ary codes of covering radius one

2000 ◽  
Vol 219 (1-3) ◽  
pp. 97-106 ◽  
Author(s):  
Wolfgang Haas
Keyword(s):  
Lower Bounds ◽  
Covering Radius

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

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