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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 ...

##
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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

1985
*
IEEE Transactions on Information Theory
*

There are

doi:10.1109/tit.1985.1057043
fatcat:bq2zv7acnbapta45kd7lkbca7u
*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. ...##
###
Application of coding theory to interconnection networks

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

doi:10.1016/0166-218x(92)90158-7
fatcat:une42obh6ngrfgdpqxzhdlmdau
*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. ...##
###
Reformulation of the covering and quantizer problems as ground states of interacting particles

2010
*
Physical Review E
*

In the case of the quantizer problem, we derive improved

doi:10.1103/physreve.82.056109
pmid:21230547
fatcat:ajbhcarrbvaptmpsr3ctqvmqwi
*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*. ...##
###
Guest Editorial Special Issue: "From Deletion-Correction to Graph Reconstruction: In Memory of Vladimir I. Levenshtein"

2021
*
IEEE Transactions on Information Theory
*

The authors present

doi:10.1109/tit.2021.3072555
dblp:journals/tit/BargDGKKMMMZ21
fatcat:ebzt3z72yjfwtbiqczqjqzwzk4
*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 ...##
###
Theoretical upper bounds on the covering radii of Boolean functions

2004
*
Journal of Discrete Mathematical Sciences and Cryptography
*

We prove new

doi:10.1080/09720529.2004.10698003
fatcat:q5p66z6o25eopao7rz2loyrh2m
*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*. ...##
###
Finite $\epsilon$-unit distance graphs

2021
*
Journal of Algebra Combinatorics Discrete Structures and Applications
*

In 2016, Grytczuk et al. proved

doi:10.13069/jacodesmath.1000784
fatcat:43sallhc6bbkbowfjfxey3w6lm
*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 . ...##
###
On Lower Bounds of the Density of Delone Sets and Holes in Sequences of Sphere Packings

2005
*
Experimental Mathematics
*

In Table 2 we summarize some values and known

doi:10.1080/10586458.2005.10128908
fatcat:oyrnn7q6w5azxlm5eg7cz5qfnq
*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 ...##
###
Book Review: Sphere packings, lattices and groups\/}, second ed

1993
*
Bulletin of the American Mathematical Society
*

The difference between the exponent in Kolmogorov's law obtained here and the

doi:10.1090/s0273-0979-1993-00435-x
fatcat:4jfjzmpphjabtebaa4fjvrs7gq
*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 ...##
###
A Spectral Approach to Lower Bounds with Applications to Geometric Searching

1998
*
SIAM journal on computing (Print)
*

We establish

doi:10.1137/s0097539794275665
fatcat:tdw3gtbb7vdolbynailm5ynd34
*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 . ...##
###
Compression for Quadratic Similarity Queries: Finite Blocklength and Practical Schemes
[article]

2014
*
arXiv
*
pre-print

As

arXiv:1404.5173v2
fatcat:5z4ac3enqbczxkfkraat5b35du
*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). ...##
###
A deterministic view on the capacity of bandlimited functions

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. ...

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

2015
*
Sbornik. Mathematics
*

We use technique, based

doi:10.1070/sm2015v206n11abeh004507
fatcat:xt3k7no4rrdkvoakbsij3pcsgq
*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. ...##
###
Covering arrays and intersecting codes

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 . ...

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