scholarly journals Graphs Associated with Codes of Covering Radius 1 and Minimum Distance 2

10.37236/792 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Joanne L. Hall
Keyword(s):  
Upper Bound ◽  
Minimum Distance ◽  
Latin Squares ◽  
Covering Radius ◽  
Unique Graph ◽  
Partial Latin Squares

The search for codes of covering radius $1$ led Östergård, Quistorff and Wassermann to the OQW method of associating a unique graph to each code. We present results on the structure and existence of OQW-associated graphs. These are used to find an upper bound on the size of a ball of radius $1$ around a code of length $3$ and minimum distance $2$. OQW-associated graphs and non-extendable partial Latin squares are used to catalogue codes of length $3$ over $4$ symbols with covering radius $1$ and minimum distance $2$.

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 A Bound on Permutation Codes

10.37236/2929 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Jürgen Bierbrauer ◽  
Klaus Metsch
Keyword(s):  
Projective Plane ◽  
Symmetric Group ◽  
Minimum Distance ◽  
Main Part ◽  
Latin Squares ◽  
Permutation Codes ◽  
Permutation Code ◽  
Mutually Orthogonal Latin Squares ◽  
Hamming Metric ◽  
Completion Theorem

Consider the symmetric group $S_n$ with the Hamming metric. A  permutation code on $n$ symbols is a subset $C\subseteq S_n.$ If $C$ has minimum distance $\geq n-1,$ then $\vert C\vert\leq n^2-n.$ Equality can be reached if and only if a projective plane of order $n$ exists. Call $C$ embeddable if it is contained in a permutation code of minimum distance $n-1$ and cardinality $n^2-n.$ Let $\delta =\delta (C)=n^2-n-\vert C\vert$ be the deficiency of the permutation code $C\subseteq S_n$ of minimum distance $\geq n-1.$We prove that $C$ is embeddable if either $\delta\leq 2$ or if $(\delta^2-1)(\delta +1)^2<27(n+2)/16.$ The main part of the proof is an adaptation of the method used to obtain the famous Bruck completion theorem for mutually orthogonal latin squares.

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.

A New Upper Bound on the Minimum Distance of Turbo Codes

2004 ◽  
Vol 50 (12) ◽  
pp. 2985-2997 ◽  
Author(s):  
A. Perotti ◽  
S. Benedetto
Keyword(s):  
Upper Bound ◽  
Turbo Codes ◽  
Minimum Distance

EVANS' NORMAL FORM THEOREM REVISITED

2007 ◽  
Vol 17 (08) ◽  
pp. 1577-1592 ◽  
Author(s):  
JONATHAN D. H. SMITH
Keyword(s):  
Normal Form ◽  
Symmetric Group ◽  
Word Problem ◽  
Latin Squares ◽  
Form Theorem ◽  
Left Regular ◽  
Normal Form Theorem ◽  
Partial Latin Squares

Evans defined quasigroups equationally, and proved a Normal Form Theorem solving the word problem for free extensions of partial Latin squares. In this paper, quasigroups are redefined as algebras with six basic operations related by triality, manifested as coupled right and left regular actions of the symmetric group on three symbols. Triality leads to considerable simplifications in the proof of Evans' Normal Form Theorem, and makes it directly applicable to each of the six major varieties of quasigroups defined by subgroups of the symmetric group. Normal form theorems for the corresponding varieties of idempotent quasigroups are obtained as immediate corollaries.

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