scholarly journals Anti-Codes in Terms of Berlekamp's Switching Game

10.37236/17 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Projective Space ◽  
Linear Code ◽  
Coding Theory ◽  
Normal Forms ◽  
Light Pattern ◽  
Maximal Weight ◽  
Axis Parallel ◽  
Generalized Games ◽  
Switching Game ◽  
Zero Points

We view a linear code (subspace) $C\leq\mathbb{F}_{q}^n$ as a light pattern on the \(n\)-dimensional Berlekamp Board $\mathbb{F}_{q}^n$ with $q^n$ light bulbs. The lights corresponding to elements of $C$ are ON, the others are OFF. Then we allow axis-parallel switches of complete rows, columns, etc. We show that the dual code $C^\perp$ contains a vector $v$ of full weight, i.e. $v_1,v_2,\dots,v_n\neq0$, if and only if the light pattern $C$ cannot be switched off. Generalizations of this allow us to describe anti-codes with maximal weight $\delta$ in a similar way, or, alternatively, in terms of a switching game in projective space. We provide convenient bases and normal forms to the modules of all light patterns of the generalized games. All our proofs are purely combinatorial and simpler than the algebraic ones used for similar results about anti-codes in $\mathbb{Z}_k^n$.  Aside from coding theory, the game is also of interest in the study of nowhere-zero points of matrices and nowhere-zero flows and colorings of graphs.

scholarly journals Colorings and Nowhere-Zero Flows of Graphs in Terms of Berlekamp's Switching Game

10.37236/552 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Normal Forms ◽  
Oriented Graph ◽  
Vertex Coloring ◽  
Cycle Space ◽  
The Core ◽  
Light Pattern ◽  
Special Cases ◽  
Axis Parallel ◽  
Switching Game ◽  
Zero Points

We work with a unifying linear algebra formulation for nowhere-zero flows and colorings of graphs and matrices. Given a subspace (code) $U\leq{\mathbb{Z}_k^n}$ – e.g. the bond or the cycle space over ${\mathbb{Z}}_k$ of an oriented graph – we call a nowhere-zero tuple $f\in{\mathbb{Z}_k^n}$ a flow of $U$ if $f$ is orthogonal to $U$. In order to detect flows, we view the subspace $U$ as a light pattern on the $n$-dimensional Berlekamp Board ${\mathbb{Z}_k^n}$ with $k^n$ light bulbs. The lights corresponding to elements of $U$ are ON, the others are OFF. Then we allow axis-parallel switches of complete rows, columns, etc. The core result of this paper is that the subspace $U$ has a flow if and only if the light pattern $U$ cannot be switched off. In particular, a graph $G$ has a nowhere-zero $k$-flow if and only if the ${\mathbb{Z}}_k$-bond space of $G$ cannot be switched off. It has a vertex coloring with $k$ colors if and only if a certain corresponding code over ${\mathbb{Z}}_k$ cannot be switched off. Similar statements hold for Tait colorings, and for nowhere-zero points of matrices. Studying different normal forms to equivalence classes of light patterns, we find various new equivalents, e.g., for the Four Color Problem, Tutte's Flow Conjectures and Jaeger's Conjecture. Two of our equivalents for colorability and existence of nowhere zero flows of graphs include as special cases results by Matiyasevich, by Balázs Szegedy, and by Onn. Alon and Tarsi's sufficient condition for $k$-colorability also arrives, remarkably, as a generalized full equivalent.

scholarly journals Counting Z2 Z4 Z8-additive Codes

2019 ◽  
Vol 12 (2) ◽  
pp. 668-679
Author(s):  
Basri Çalışkan ◽  
Kemal Balıkçı
Keyword(s):  
Linear Code ◽  
Coding Theory ◽  
Linear Codes ◽  
Algebraic Coding Theory ◽  
Additive Codes ◽  
Generator Matrices

In Algebraic Coding Theory, all linear codes are described by generator matrices. Any linear code has many generator matrices which are equivalent. It is important to find the number of the generator matrices for constructing of these codes. In this paper, we study Z_2 Z_4 Z_8-additive codes, which are the extension of recently introduced Z_2 Z_4-additive codes. We count the number of arbitrary Z_2 Z_4 Z_8-additive codes. Then we investigate connections to Z_2 Z_4 and Z_2 Z_8-additive codes with Z_2 Z_4 Z_8, and give some illustrative examples.

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 Nonexistence of some ternary linear codes with minimum weight -2 modulo 9

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Toshiharu Sawashima ◽  
Tatsuya Maruta
Keyword(s):  
Linear Code ◽  
Coding Theory ◽  
Linear Codes ◽  
Minimum Weight ◽  
Extension Theorem ◽  
Minimum Length ◽  
Geometric Methods ◽  
Optimal Linear ◽  
Ternary Linear Codes ◽  
Optimal Linear Codes

<p style='text-indent:20px;'>One of the fundamental problems in coding theory is to find <inline-formula><tex-math id="M3">\begin{document}$ n_q(k,d) $\end{document}</tex-math></inline-formula>, the minimum length <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> for which a linear code of length <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>, dimension <inline-formula><tex-math id="M6">\begin{document}$ k $\end{document}</tex-math></inline-formula>, and the minimum weight <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> over the field of order <inline-formula><tex-math id="M8">\begin{document}$ q $\end{document}</tex-math></inline-formula> exists. The problem of determining the values of <inline-formula><tex-math id="M9">\begin{document}$ n_q(k,d) $\end{document}</tex-math></inline-formula> is known as the optimal linear codes problem. Using the geometric methods through projective geometry and a new extension theorem given by Kanda (2020), we determine <inline-formula><tex-math id="M10">\begin{document}$ n_3(6,d) $\end{document}</tex-math></inline-formula> for some values of <inline-formula><tex-math id="M11">\begin{document}$ d $\end{document}</tex-math></inline-formula> by proving the nonexistence of linear codes with certain parameters.</p>

The limits of strain and lattice parameter measurements by CBED

1989 ◽  
Vol 47 ◽  
pp. 518-519
Author(s):  
Hamish L. Fraser
Keyword(s):  
Electron Beam ◽  
Mirror Symmetry ◽  
Local Symmetry ◽  
Lattice Parameter ◽  
Growth Direction ◽  
Surface Relaxation ◽  
Strained Layer ◽  
Axis Parallel ◽  
Local Lattice ◽  
Strained Layer Superlattice

The topic of strain and lattice parameter measurements using CBED is discussed by reference to several examples. In this paper, only one of these examples is referenced because of the limitation of length. In this technique, scattering in the higher order Laue zones is used to determine local lattice parameters. Work (e.g. 1) has concentrated on a model strained-layer superlattice, namely Si/Gex-Si1-x. In bulk samples, the strain is expected to be tetragonal in nature with the unique axis parallel to [100], the growth direction. When CBED patterns are recorded from the alloy epi-layers, the symmetries exhibited by the patterns are not tetragonal, but are in fact distorted from this to lower symmetries. The spatial variation of the distortion close to a strained-layer interface has been assessed. This is most readily noted by consideration of Fig. 1(a-c), which show enlargements of CBED patterns for various locations and compositions of Ge. Thus, Fig. 1(a) was obtained with the electron beam positioned in the center of a 5Ge epilayer and the distortion is consistent with an orthorhombic distortion. When the beam is situated at about 150 nm from the interface, the same part of the CBED pattern is shown in Fig. 1(b); clearly, the symmetry exhibited by the mirror planes in Fig. 1 is broken. Finally, when the electron beam is positioned in the center of a 10Ge epilayer, the CBED pattern yields the result shown in Fig. 1(c). In this case, the break in the mirror symmetry is independent of distance form the heterointerface, as might be expected from the increase in the mismatch between 5 and 10%Ge, i.e. 0.2 to 0.4%, respectively. From computer simulation, Fig.2, the apparent monocline distortion corresponding to the 5Ge epilayer is quantified as a100 = 0.5443 nm, a010 = 0.5429 nm and a001 = 0.5440 nm (all ± 0.0001 nm), and α = β = 90°, γ = 89.96 ± 0.02°. These local symmetry changes are most likely due to surface relaxation phenomena.

Normal Forms and Bifurcation of Planar Vector Fields

1994 ◽  
Author(s):  
Shui-Nee Chow ◽  
Chengzhi Li ◽  
Duo Wang
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Planar Vector Fields ◽  
Planar Vector

Coding Theory

2004 ◽  
Author(s):  
San Ling ◽  
Chaoping Xing
Keyword(s):  
Coding Theory

Graph Theory, Coding Theory and Block Designs

1975 ◽  
Author(s):  
P. J. Cameron ◽  
J. H. van Lint
Keyword(s):  
Graph Theory ◽  
Coding Theory ◽  
Block Designs

The sense of presence in virtual environments:

2003 ◽  
Vol 15 (2) ◽  
pp. 69-71 ◽  
Author(s):  
Thomas W. Schubert
Keyword(s):  
Virtual Environments ◽  
Coding Theory ◽  
Model Building ◽  
Self Report ◽  
Survey Study ◽  
Spatial Presence ◽  
Component Structure ◽  
Sense Of Presence ◽  
Being There ◽  
Measure Sense

Abstract. The sense of presence is the feeling of being there in a virtual environment. A three-component self report scale to measure sense of presence is described, the components being sense of spatial presence, involvement, and realness. This three-component structure was developed in a survey study with players of 3D games (N = 246) and replicated in a second survey study (N = 296); studies using the scale for measuring the effects of interaction on presence provide evidence for validity. The findings are explained by the Potential Action Coding Theory of presence, which assumes that presence develops from mental model building and suppression of the real environment.

Realizing Imagery: A Retrospective Examination of Dual Coding Theory

1996 ◽  
Vol 41 (6) ◽  
pp. 587-588
Author(s):  
Valerie A. Thompson
Keyword(s):  
Coding Theory ◽  
Dual Coding ◽  
Dual Coding Theory ◽  
Retrospective Examination
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