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Page 4839 of Mathematical Reviews Vol. , Issue 84k [page]

1984 Mathematical Reviews  
A nonconstructive upper bound on covering radius. IEEE Trans. Inform. Theory 29 (1983), no. 3, 352-353. Author’s summary: “Let ¢(n,k) denote the minimum covering radius of a binary linear (n,k) code.  ...  We give a nonconstructive upper bound on t(n,k), which coincides asymptotically with the known lower bound, namely n-'t(n,nR)=H~'(1—R)+O(n"' logn), where R is fixed, 0<R<1, and H™' is the inverse of the  ... 

Page 5928 of Mathematical Reviews Vol. , Issue 86m [page]

1986 Mathematical Reviews  
Author summary: “We give a nonconstructive proof of the exis- tence of good coverings of binary and non-binary Hamming spaces by spheres centered on a subspace (linear codes).  ...  There is an asymptotic upper bound due to S. G. Vladdut and V. G. Drinfel’d [Functional Anal.  ... 

Covering radius---Survey and recent results

G. Cohen, M. Karpovsky, H. Mattson, J. Schatz
1985 IEEE Transactions on Information Theory  
There are a number of upper and lower bounds, including asymptotic results, a few exact determinations of covering radius, some extensive relations with other aspects of coding theory through the Reed-Muller  ...  There is also a recent result on the complexity of computing the covering radius.  ...  UPPER BOUNDS ON COVERING RADIUS A. The Redundancy and Delsurte Bounds The first and simplest bound on covering radius is given by the following proposition.  ... 
doi:10.1109/tit.1985.1057043 fatcat:bq2zv7acnbapta45kd7lkbca7u

Application of coding theory to interconnection networks

Gilles Zémor, Gérard D. Cohen
1992 Discrete Applied Mathematics  
These concepts have coding-theoretical counterparts in several situations: for instance, Cayley graphs can be associated with codes to yield the following correspondences: diameter +P covering radius,  ...  The aim of this paper is to show how coding theory can interpret some networktheoretical problems, to use the correspondence to give bounds on the parameters of an interconnection network, and to suggest  ...  Fellows for many valuable comments on a first version of this work.  ... 
doi:10.1016/0166-218x(92)90158-7 fatcat:une42obh6ngrfgdpqxzhdlmdau

Reformulation of the covering and quantizer problems as ground states of interacting particles

S. Torquato
2010 Physical Review E  
In the case of the quantizer problem, we derive improved upper bounds on the quantizer error using sphere-packing solutions, which are generally substantially sharper than an existing upper bound in low  ...  Finally, we remark on possible applications of our results for the detection of gravitational waves.  ...  Rogers showed that (possibly nonlattice) coverings exist with θ ≤ d ln(d) + d ln(ln(d)) + 5d. (42) for d ≥ 2. This is a nonconstructive upper bound.  ... 
doi:10.1103/physreve.82.056109 pmid:21230547 fatcat:ajbhcarrbvaptmpsr3ctqvmqwi

Guest Editorial Special Issue: "From Deletion-Correction to Graph Reconstruction: In Memory of Vladimir I. Levenshtein"

Alexander Barg, Lara Dolecek, Ryan Gabrys, Gyula O. H. Katona, János Körner, Andrew McGregor, Olgica Milenkovic, Sihem Mesnager, Gilles Zémor
2021 IEEE Transactions on Information Theory  
The authors present upper and lower bounds and show that the asymptotic rate of the resulting codes matches the best-known bounds for covering codes under the Hamming distance.  ...  the Plotkin bound [item 6) in the Appendix]; derived upper-bounds on the size of comma-free codes and proposed explicit constructions for the same [item 7) in the Appendix]; and derived new bounds for  ... 
doi:10.1109/tit.2021.3072555 dblp:journals/tit/BargDGKKMMMZ21 fatcat:ebzt3z72yjfwtbiqczqjqzwzk4

Theoretical upper bounds on the covering radii of Boolean functions

Michel Mitton
2004 Journal of Discrete Mathematical Sciences and Cryptography  
We prove new upper bounds for the covering radii ρ(n) and ρ B (n) of the first order Reed-Muller code R(1, n).  ...  Although these bounds be actually theoretical, they improve the classical Helleseth-Kløve-Mykkeltveit (H.K.M.) bound 2 n−1 − 2 n 2 −1 .  ...  Conclusion We have obtained theoretical upper bounds (7),(8) on ρ(n) and ρ B (n). Except the already known ρ(n) for even n, theses bounds minorate the H.K.M. bound.  ... 
doi:10.1080/09720529.2004.10698003 fatcat:q5p66z6o25eopao7rz2loyrh2m

Finite $\epsilon$-unit distance graphs

2021 Journal of Algebra Combinatorics Discrete Structures and Applications  
In 2016, Grytczuk et al. proved a weaker result with a human-comprehensible but nonconstructive proof: whenever 0 < < 1, we have that χ(G ) ≥ 5.  ...  In a 2018 preprint, de Grey shows that χ(G0) ≥ 5; the proof relies heavily on machine computation.  ...  Cover A with finitely many open disks of radius /7, each centered at a point of A. For each disk in this covering, choose an element of V .  ... 
doi:10.13069/jacodesmath.1000784 fatcat:43sallhc6bbkbowfjfxey3w6lm

On Lower Bounds of the Density of Delone Sets and Holes in Sequences of Sphere Packings

G. Muraz, J. -L. Verger-Gaugry
2005 Experimental Mathematics  
In Table 2 we summarize some values and known upper bounds of R L (n). The following theorem is fundamental but nonconstructive. Theorem 3.1. (Butler.)  ...  TABLE 1 . 1 Kabatjanskiȋ and Levenstein [Kabatjanskiȋ and Levenstein 78] Upper bounds of δ as a function of n. 29] Table 1 , 1 the best one being the one of Kabatjanskiȋ and Levenstein ([Rogers 64  ... 
doi:10.1080/10586458.2005.10128908 fatcat:oyrnn7q6w5azxlm5eg7cz5qfnq

Book Review: Sphere packings, lattices and groups\/}, second ed

C. A. Rogers
1993 Bulletin of the American Mathematical Society  
The difference between the exponent in Kolmogorov's law obtained here and the one obtained in a similar way by Kraichnan [1] is not explained; Kraichnan found that to recover Kolmogorov's result he needed  ...  The present book may be a useful guide to the Soviet literature for experts in this field, but it is hard to recommend to a broader audience.  ...  The second edition differs from the first by the inclusion of a survey of a surprisingly large body of work that has been completed since the first edition appeared and the addition of a supplementary  ... 
doi:10.1090/s0273-0979-1993-00435-x fatcat:4jfjzmpphjabtebaa4fjvrs7gq

A Spectral Approach to Lower Bounds with Applications to Geometric Searching

Bernard Chazelle
1998 SIAM journal on computing (Print)  
We establish a nonlinear lower bound for halfplane range searching over a group.  ...  This is the first nontrivial lower bound for range searching over a group. By contrast, range searching over a semigroup (which forbids subtractions) is almost completely understood.  ...  Within a factor of 2 m+1 , the total number of such points is an upper bound on N .  ... 
doi:10.1137/s0097539794275665 fatcat:tdw3gtbb7vdolbynailm5ynd34

Compression for Quadratic Similarity Queries: Finite Blocklength and Practical Schemes [article]

Fabian Steiner, Steffen Dempfle, Amir Ingber, Tsachy Weissman
2014 arXiv   pre-print
As a side result, we obtain a bound on the covering angle of any wrapped spherical code, as a function of the covering radius of the underlying lattice.  ...  We then give a nonasymptotic lower bound on the performance of any compression scheme, and compare to the upper (achievability) bound.  ...  Theorem 6 (Upper Bound on the Covering Angle).  ... 
arXiv:1404.5173v2 fatcat:5z4ac3enqbczxkfkraat5b35du

A deterministic view on the capacity of bandlimited functions

Taehyung J. Lim, Massimo Franceschcetti
2014 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton)  
Upper and lower bounds on the Kolmogorov capacity are derived, that improve upon previous results.  ...  The functional form of these bounds is analogous to the one of the capacity of the additive white Gaussian noise channel, showing an essential similarity between the deterministic and the stochastic approaches  ...  Upper Bound We now derive an upper bound on the 2 -capacity of the space of bandlimited functions.  ... 
doi:10.1109/allerton.2014.7028522 dblp:conf/allerton/LimF14 fatcat:6emaka2fr5ejddz72xevkuulhq

Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness

V N Temlyakov
2015 Sbornik. Mathematics  
We use technique, based on a combination of results from hyperbolic cross approximation, which were obtained in 1980s -- 1990s, and recent results on greedy approximation to obtain sharp estimates for  ...  We prove lower bounds, which show that one cannot improve accuracy of sparse grids methods with 2^nn^d-1 points in the grid by adding 2^n arbitrary points.  ...  The upper bounds are provided by a constructive method A m (·, ∞, µ) based on greedy algorithms. Consider the case σ m (W r 1,α ) p , which is not covered by Theorems 2.7 and 2.8.  ... 
doi:10.1070/sm2015v206n11abeh004507 fatcat:xt3k7no4rrdkvoakbsij3pcsgq

Covering arrays and intersecting codes

N. J. A. Sloane
1993 Journal of combinatorial designs (Print)  
A t-covering array is a set of k binary vectors of length n with the property that, in any t coordinate positions, all 2t possibilities occur at least once.  ...  This article studies the properties of 3-covering arrays and intersecting codes, and gives a table of the best 3-covering arrays presently known.  ...  upper bounds presently known on f 2 ( K ) for small values of K .  ... 
doi:10.1002/jcd.3180010106 fatcat:fzaevcbqsrbzfp6gvgcxswfzpm
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