New upper bounds on the expected length of one-to-one codes

2003 ◽  
Author(s):  
C. Weidmann
Keyword(s):  
Upper Bounds ◽  
Expected Length ◽  
One To One

New Bounds on the Expected Length of Optimal One-to-One Codes

2007 ◽  
Vol 53 (5) ◽  
pp. 1884-1895 ◽  
Author(s):  
Jay Cheng ◽  
Tien-Ke Huang ◽  
Claudio Weidmann
Keyword(s):  
Expected Length ◽  
One To One

New bounds on the expected length of one-to-one codes

1996 ◽  
Vol 42 (1) ◽  
pp. 246-250 ◽  
Author(s):  
C. Blundo ◽  
R. De Prisco
Keyword(s):  
Expected Length ◽  
One To One

scholarly journals Upper bounds on the length function for covering codes with covering radius $ R $ and codimension $ tR+1 $

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Alexander A. Davydov ◽  
Stefano Marcugini ◽  
Fernanda Pambianco
Keyword(s):  
Projective Space ◽  
Linear Code ◽  
Prime Power ◽  
Upper Bounds ◽  
Covering Radius ◽  
Length Function ◽  
Covering Codes ◽  
New Construction ◽  
One To One ◽  
Better Than

<p style='text-indent:20px;'>The length function <inline-formula><tex-math id="M3">\begin{document}$ \ell_q(r,R) $\end{document}</tex-math></inline-formula> is the smallest length of a <inline-formula><tex-math id="M4">\begin{document}$ q $\end{document}</tex-math></inline-formula>-ary linear code with codimension (redundancy) <inline-formula><tex-math id="M5">\begin{document}$ r $\end{document}</tex-math></inline-formula> and covering radius <inline-formula><tex-math id="M6">\begin{document}$ R $\end{document}</tex-math></inline-formula>. In this work, new upper bounds on <inline-formula><tex-math id="M7">\begin{document}$ \ell_q(tR+1,R) $\end{document}</tex-math></inline-formula> are obtained in the following forms:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{split} &amp;(a)\; \ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &amp;\phantom{(a)\; } q\;{\rm{ is \;an\; arbitrary \;prime\; power}},\; c{\rm{ \;is\; independent \;of\; }}q. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \begin{split} &amp;(b)\; \ell_q(r,R)&lt; 3.43Rq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &amp;\phantom{(b)\; } q\;{\rm{ is \;an\; arbitrary\; prime \;power}},\; q\;{\rm{ is \;large\; enough}}. \end{split} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>In the literature, for <inline-formula><tex-math id="M8">\begin{document}$ q = (q')^R $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M9">\begin{document}$ q' $\end{document}</tex-math></inline-formula> a prime power, smaller upper bounds are known; however, when <inline-formula><tex-math id="M10">\begin{document}$ q $\end{document}</tex-math></inline-formula> is an arbitrary prime power, the bounds of this paper are better than the known ones.</p><p style='text-indent:20px;'>For <inline-formula><tex-math id="M11">\begin{document}$ t = 1 $\end{document}</tex-math></inline-formula>, we use a one-to-one correspondence between <inline-formula><tex-math id="M12">\begin{document}$ [n,n-(R+1)]_qR $\end{document}</tex-math></inline-formula> codes and <inline-formula><tex-math id="M13">\begin{document}$ (R-1) $\end{document}</tex-math></inline-formula>-saturating <inline-formula><tex-math id="M14">\begin{document}$ n $\end{document}</tex-math></inline-formula>-sets in the projective space <inline-formula><tex-math id="M15">\begin{document}$ \mathrm{PG}(R,q) $\end{document}</tex-math></inline-formula>. A new construction of such saturating sets providing sets of small size is proposed. Then the <inline-formula><tex-math id="M16">\begin{document}$ [n,n-(R+1)]_qR $\end{document}</tex-math></inline-formula> codes, obtained by geometrical methods, are taken as the starting ones in the lift-constructions (so-called "<inline-formula><tex-math id="M17">\begin{document}$ q^m $\end{document}</tex-math></inline-formula>-concatenating constructions") for covering codes to obtain infinite families of codes with growing codimension <inline-formula><tex-math id="M18">\begin{document}$ r = tR+1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M19">\begin{document}$ t\ge1 $\end{document}</tex-math></inline-formula>.</p>

A lower bound on the expected length of one-to-one codes

1994 ◽  
Vol 40 (5) ◽  
pp. 1670-1672 ◽  
Author(s):  
N. Alon ◽  
A. Orlitsky
Keyword(s):  
Lower Bound ◽  
Expected Length ◽  
One To One

scholarly journals Short Score Certificates for Upset Tournaments

10.37236/1362 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Jeffrey L. Poet ◽  
Bryan L. Shader
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
General Construction ◽  
Transitive Tournament ◽  
One To One

A score certificate for a tournament, $T$, is a collection of arcs of $T$ which can be uniquely completed to a tournament with the same score-list as $T$'s, and the score certificate number of $T$ is the least number of arcs in a score certificate of $T$. Upper bounds on the score certificate number of upset tournaments are derived. The upset tournaments on $n$ vertices are in one–to–one correspondence with the ordered partitions of $n-3$, and are "almost" transitive tournaments. For each upset tournament on $n$ vertices a general construction of a score certificate with at most $2n-3$ arcs is given. Also, for the upset tournament, $T_{\lambda}$, corresponding to the ordered partition $\lambda$, a score certificate with at most $n+2k+3$ arcs is constructed, where $k$ is the number of parts of $\lambda$ of size at least 2. Lower bounds on the score certificate number of $T_{\lambda}$ in the case that each part is sufficiently large are derived. In particular, the score certificate number of the so-called nearly transitive tournament on $n$ vertices is shown to be $n+3$, for $n\geq 10$.

Sign in / Sign up

Export Citation Format

Share Document