New Upper Bounds on the Redundancy of Optimal One-to-One Codes

2007 ◽  
Author(s):  
Tien-Ke Huang ◽  
Jay Cheng ◽  
Chin-Liang Wang
Keyword(s):  
Upper Bounds ◽  
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>

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 $n\geq 10$.

A Support Program

1994 ◽  
Vol 25 (2) ◽  
pp. 112-114 ◽  
Author(s):  
Henna Grunblatt ◽  
Lisa Daar
Keyword(s):  
Hearing Aids ◽  
School Counselor ◽  
Support Program ◽  
Educational Audiology ◽  
Part Program ◽  
One To One

A program for providing information to children who are deaf about their deafness and addressing common concerns about deafness is detailed. Developed by a school audiologist and the school counselor, this two-part program is geared for children from 3 years to 15 years of age. The first part is an educational audiology program consisting of varied informational classes conducted by the audiologist. Five topics are addressed in this part of the program, including basic audiology, hearing aids, FM systems, audiograms, and student concerns. The second part of the program consists of individualized counseling. This involves both one-to-one counseling sessions between a student and the school counselor, as well as conjoint sessions conducted—with the student’s permission—by both the audiologist and the school counselor.

Review of One to One: The Experience of Psychotherapy.

1989 ◽  
Vol 34 (10) ◽  
pp. 958-958
Author(s):  
No authorship indicated
Keyword(s):  
One To One
Sign in / Sign up

Export Citation Format

Share Document