Lattices and Spherical Codes

Costa, Sueli I. R.; Oggier, Frédérique; Campello, Antonio; Belfiore, Jean-Claude; Viterbo, Emanuele

doi:10.1007/978-3-319-67882-5_5

Cited by 1 publication

(2 citation statements)

References 29 publications

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“…A count is given if some of the parameters were resolved. [30,20] 50014 N 1/r [31,18] 6049 N 1/r [33,32] 50014 N 1/r [34,16] 50014 N 1/r [37,6] 10018 N 1/r [38,3] 50014 N 1/r [39,10] 50014 N 1/r [40,9] 50014 N 1/r [41,10] 50014 N 1/r [42,5] 50014 N 1/r [43,21] 50014 N 1/r [44,3] 50014 N 1/r [45,14] 50014 N 1/r [46,24] 37279 N 1/r [48,18] 1021 N 1/r [49,32] 28205 N 1/r [50,4] 50014 N 1/r [51,16] 50014 N 1/r [52,34] 9921 N 1/r [53,27] 1001 N 1/r [57,18] 50014 N 1/r [60,30] There are 6 unknown spherical codes in these two points. 47 points requires about 47 parameters, and 58 points requires 84 parameters.…”

Section: Summary Of Algebraic Resultsmentioning

confidence: 99%

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Some spherical codes in S2 and their algebraic numbers

Rathbun¹,

Ridgway²

2020

Preprint

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The first 195 spherical codes for the global minima of 1-65 points on S2 have been obtained for 3 types of potentials: logarithmic −log(r), Coulomb −1/r, called the Thomson problem, and the inverse square law −1/r 2 , with 77, 38, and 38 digits precision respectively.It was discovered that certain point sets have embedded polygonal structures, constraining the points, enabling them to be parameterized and to successfully recover the algebraic polynomial.So far 49 algebraic number sets have been recovered, but 109 more remain to be recovered from their 1,622 parameters, 983 known to 50,014 digit precision. The very high algebraic degree of these minimal polynomials eludes finding the algebraic numbers from the spherical codes and requires new mathematical tools to meet this challenge.

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Section: Summary Of Algebraic Resultsmentioning

confidence: 99%

“…The decimal values of the 10 parameters, a − j, and energy are known to 50,014 digits, nevertheless, the algebraic polynomials were not obtained, they are > 360 in degree. Gram groups [ [2,1], [6,6], [12,31], [18,1], [24,49], [36,1], [41,1]]…”

Section: Minimal Energy Values -mentioning

confidence: 99%

Some spherical codes in S2 and their algebraic numbers

Rathbun¹,

Ridgway²

2020

Preprint

View full text Add to dashboard Cite

show abstract

Lattices and Spherical Codes

Cited by 1 publication

References 29 publications

Some spherical codes in S2 and their algebraic numbers

Some spherical codes in S2 and their algebraic numbers

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