The covering radius of the Leech lattice

1982 ◽  
Vol 380 (1779) ◽  
pp. 261-290 ◽  
Keyword(s):  
Minimum Distance ◽  
Dimensional Space ◽  
Covering Radius ◽  
Lattice Points ◽  
Maximum Distance ◽  
Leech Lattice

We investigate the points in 24-dimensional space at maximum distance from the Leech lattice, i. e. the ‘deepest holes’ in that lattice. The maximum distance of any such point from the Leech lattice is shown to be 1/√2 times the minimum distance between the lattice points. Furthermore there are 23 types of ‘deepest hole’, one for each of the 23 even unimodular 24-dimensional lattices found by Niemeier.

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 Comparison of the internal fit of metal crowns fabricated by traditional casting, computer numerical control milling, and three-dimensional printing

PLoS ONE ◽  
2021 ◽  
Vol 16 (9) ◽  
pp. e0257158
Author(s):  
Wei-Ting Chou ◽  
Chuan-Chung Chuang ◽  
Yi-Bing Wang ◽  
Hsien-Chung Chiu
Keyword(s):  
3D Printing ◽  
Minimum Distance ◽  
Statistical Significance ◽  
Three Dimensional ◽  
Reference Points ◽  
Numerical Control ◽  
Maximum Distance ◽  
Cnc Milling ◽  
Low Viscosity ◽  
Internal Fit

This experimental study aimed to compare the internal fit (marginal fit and internal discrepancy) of metal crowns fabricated by traditional casting and digital methods (computer numerically controlled (CNC) milling and three-dimensional [3D] printing). Thirty standard master abutment models were fabricated using a 3D printing technique with digital software. Metal crowns were fabricated by traditional casting, CNC milling, and 3D printing. The silicon replica method was used to measure the marginal and internal fit. A thin layer of low-viscosity polyvinyl siloxane material was placed inside each crown and on the die (like a seat) until the material was set. Replicas were examined at four reference points under a microscope: the central pit (M1), cusp tip (M2), axial wall (M3), and margin (M4). The measured data were analyzed using a one-way analysis of variance (ANOVA) to verify statistical significance, which was set at p < 0.05. In the traditional casting group, the minimum distance measured was at M3 (90.68 ± 14.4 μm) and the maximum distance measured was at M1 (145.12 ± 22 μm). In the milling group, the minimum distance measured was at M3 (71.85 ± 23.69 μm) and the maximum distance measured was at M1 (108.68 ± 10.52 μm). In the 3D printing group, the minimum distance measured was at M3 (100.59 ± 9.26 μm) and the maximum distance measured was at M1 (122.33 ± 7.66 μm). The mean discrepancy for the traditional casting, CNC milling, and 3D printing groups was 120.20, 92.15, and 111.85 μm, respectively, showing significant differences (P < 0.05). All three methods of metal crown fabrication, that is, traditional casting, CNC milling, and 3D printing, had values within the clinically acceptable range. The marginal and internal fit of the crown was far superior in the CNC milling method.

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.

Symmetry-adapted digital modeling II. The double-helix B-DNA

2016 ◽  
Vol 72 (3) ◽  
pp. 312-323 ◽  
Author(s):  
A. Janner
Keyword(s):  
Dimensional Space ◽  
Double Helix ◽  
Three Dimensional ◽  
Reduction Factor ◽  
Lattice Points ◽  
Three Dimensions ◽  
Coarse Grained ◽  
Digital Modeling ◽  
Dna Strands ◽  
Dna Double Helix

The positions of phosphorus in B-DNA have the remarkable property of occurring (in axial projection) at well defined points in the three-dimensional space of a projected five-dimensional decagonal lattice, subdividing according to the golden mean ratio τ:1:τ [with τ = (1+\sqrt {5})/2] the edges of an enclosing decagon. The corresponding planar integral indicesn1,n2,n3,n4(which are lattice point coordinates) are extended to include the axial indexn5as well, defined for each P position of the double helix with respect to the single decagonal lattice ΛP(aP,cP) withaP= 2.222 Å andcP= 0.676 Å. A finer decagonal lattice Λ(a,c), witha=aP/6 andc=cP, together with a selection of lattice points for each nucleotide with a given indexed P position (so as to define a discrete set in three dimensions) permits the indexing of the atomic positions of the B-DNA d(AGTCAGTCAG) derived by M. J. P. van Dongen. This is done for both DNA strands and the single lattice Λ. Considered first is the sugar–phosphate subsystem, and then each nucleobase guanine, adenine, cytosine and thymine. One gets in this way a digital modeling of d(AGTCAGTCAG) in a one-to-one correspondence between atomic and indexed positions and a maximal deviation of about 0.6 Å (for the value of the lattice parameters given above). It is shown how to get a digital modeling of the B-DNA double helix for any given code. Finally, a short discussion indicates how this procedure can be extended to derive coarse-grained B-DNA models. An example is given with a reduction factor of about 2 in the number of atomic positions. A few remarks about the wider interest of this investigation and possible future developments conclude the paper.

Abelian groups, graphs and generalized knights

1959 ◽  
Vol 55 (3) ◽  
pp. 232-238 ◽  
Author(s):  
C. St J. A. Nash-Williams
Keyword(s):  
Abelian Group ◽  
Infinite Sequence ◽  
Abelian Groups ◽  
Dimensional Space ◽  
Cardinal Number ◽  
Finite Sequence ◽  
Lattice Points ◽  
The One ◽  
Infinite Abelian Group

ABSTRACTIf g is a set of generatore of an enumerably infinite Abelian group A, it is proved that the elements of A can be arranged in both a one-ended and an endless infinite sequence in which successive terms differ by ± an element of g, except that the one-ended arrangement is impossible if g is finite and the rank of A is 1. Let ν be a cardinal number. Consider an infinite ‘chessboard’ whose positions are those lattice points of ν-dimensional space which have only finitely many non-zero coordinates. A piece associated with this chessboard is a generalized knight if every vector obtainable from a move of the piece by permuting its components and changing the signs of a subset of them is itself a permitted move. It is ascertained which positions a given generalized knight can reach in a finite sequence of moves starting at the origin, and whether or not, if it can trace out the whole chessboard in (i) a one-ended, (ii) an endless infinite sequence of moves visiting each position exactly once.

A bound for the covering radius of the Leech lattice

1982 ◽  
Vol 380 (1779) ◽  
pp. 259-260 ◽  
Keyword(s):  
Covering Radius ◽  
Leech Lattice ◽  
True Value

In this paper a bound is obtained for the covering radius of the Leech lattice that is close to the subsequently obtained true value, by a method which may have more general use.

(N-Phenylimino)bis[phosphonic bis(diphenylamide)]

2007 ◽  
Vol 63 (11) ◽  
pp. o4428-o4428
Author(s):  
Daniel Chartrand ◽  
Garry S. Hanan
Keyword(s):  
Hydrogen Bond ◽  
Title Compound ◽  
Minimum Distance ◽  
Side Product ◽  
Maximum Distance ◽  
Sulfur Silicon ◽  
Hydrogen Bond Interactions

The title compound, NC6H5[PO(NHC6H5)2]2 or C30H29N5O2P2, was obtained as a side product during the addition of aniline to an amidoyl chloride, using PCl5 as chlorinating agent. The title compound was first synthesized by Murray & Woodward [(1989). Phosphorus Sulfur Silicon, 41, 399–403], again as a by-product, but no crystallographic evidence was given. The title compound crystallizes as two crystallographically unique molecules that form layers in the ab plane through O...H—N hydrogen-bond interactions, with an O...N minimum distance of 2.769 (2)Å and a maximum distance of 3.117 (2) Å. There is also one intramolecular O...H—N bond present in each of the two molecules, with an average O...N distance of 2.959 (4) Å.

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