Upper bounds on the minimum distance of spherical codes

Boyvalenkov, Peter; Danev, Danyo; Bumova, S.P.

doi:10.1109/18.532903

Cited by 36 publications

(48 citation statements)

References 18 publications

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“…Let us consider a sufficiently small turn of the facet y 1 y 2 y 3 around y 1 y 3 . If e 0 / ∈ y 1 y 3 , then this turn decreases either θ 4 (if e 0 ∈ y 1 y 2 y 3 ) or θ 2 , a contradiction. In the case e 0 ∈ y 1 y 3 any turn of y 2 around y 1 y 3 decreases φ 2,4 and does not change θ 2 .…”

Section: Corollary 4 If N > 3 Then δ 4 Is a Regular Tetrahedron Of mentioning

confidence: 96%

“…There are two cases for p k (a) (see Fig. 7): p 3 (a) = p * (a) =F {1,2} , and p 4 (a) is the intersection in H + of the great circle passing through y 1 , y 4 , and the circleS(y 1 , a 1 ) of center y 1 and radius a 1 (F {1} ⊂ S(y 1 , a 1 )). The same holds for all dimensions.…”

Section: Now We Consider the Minimum Of θmentioning

confidence: 99%

“…The inequality (3.3) plays a crucial role in the Delsarte method (see details in [2], [3], [4], [9], [13], [14], [21], [23], [27]). If z = 1/2 and…”

Section: -Cmentioning

confidence: 99%

“…In the case n = 3, G (n) k are Legendre polynomials P k , and G (4) k are Chebyshev polynomials of the second kind (but with a different normalization than usual, U k (1) = 1),…”

Section: -B the Gegenbauer Polynomials Let Us Recall Definitions Omentioning

confidence: 99%

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The kissing number in four dimensions

Musin¹

2008

Ann. Math.

120

129

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The kissing number problem asks for the maximal number k(n) of equal size nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Schütte and van der Waerden.In this paper we present a solution of a long-standing problem about the kissing number in four dimensions. Namely, the equality k(4) = 24 is proved. The proof is based on a modification of Delsarte's method.

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Section: Corollary 4 If N > 3 Then δ 4 Is a Regular Tetrahedron Of mentioning

confidence: 96%

Section: Now We Consider the Minimum Of θmentioning

confidence: 99%

“…The inequality (3.3) plays a crucial role in the Delsarte method (see details in [2], [3], [4], [9], [13], [14], [21], [23], [27]). If z = 1/2 and…”

Section: -Cmentioning

confidence: 99%

“…In the case n = 3, G (n) k are Legendre polynomials P k , and G (4) k are Chebyshev polynomials of the second kind (but with a different normalization than usual, U k (1) = 1),…”

Section: -B the Gegenbauer Polynomials Let Us Recall Definitions Omentioning

confidence: 99%

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The kissing number in four dimensions

Musin¹

2008

Ann. Math.

120

129

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“…It is also known that for both binary and non-binary sequences in the periodic sequence field, the sequences can not obtain ideal impulsive autocorrelation function (ACF) and ideal zero cross-correlation functions (CCF) simultaneously although ideal CCFs could be achieved alone. Since the ACF and CCF have to be limited by certain bounds, such as Welch bound [12], Sidelnikov bound [13], Sarwate bound [14], Levenshtein bound [15], etc. As a reulst, the concept of Zero Correlation Zone (ZCZ) [16][17] [18] during which ideal impulsive autocorrelation function and ideal zero cross-correlation functions could be achieved simultaneously is proposed to overcome the above problems.…”

Section: Introductionmentioning

confidence: 99%

Radar Sensor Network Using a Set of New Ternary Codes: Theory and Application

Liang

2011

IEEE Sensors J.

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In the radar sensor network (RSN), the interferences among different radar sensors can be effectively reduced when waveforms are properly designed. In this paper, we perform some theoretical studies on coexistence of phase coded waveforms in the RSN. We propose a new ternary codes-optimized punctured Zero Correlation Zone sequence-Pair Set (ZCZPS) and analyze their properties. Applying the new ternary codes and equal gain combination technique to the RSN, we study the detection performance versus different number of radar sensors under the different conditions. The simulation results show that the RSN using our optimized punctured ZCZPS performs better than the RSN using the same number of common codes such as the Gold codes, and much better than the single radar system no matter whether the Doppler shift is considered or not.

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On maximal Spherical codes II

1999

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We consider spherical codes attaining the Levenshtein upper bounds on the cardinality of codes with prescribed maximal inner product. We prove that the even Levenshtein bounds can be attained only by codes which are tight spherical designs. For every fixed n ≥ 5, there exist only a finite number of codes attaining the odd bounds. We derive different expressions for the distance distribution of a maximal code. As a by‐product, we obtain a result about its inner products. We describe the parameters of those codes meeting the third Levenshtein bound, which have a regular simplex as a derived code. Finally, we discuss a connection between the maximal codes attaining the third bound and strongly regular graphs. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 316–326, 1999

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Upper bounds on the minimum distance of spherical codes

Cited by 36 publications

References 18 publications

The kissing number in four dimensions

The kissing number in four dimensions

Radar Sensor Network Using a Set of New Ternary Codes: Theory and Application

On maximal Spherical codes II

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