Lee Polynomials of Codes and Theta Functions of Lattices

Several authors [2;3;10;12] have noticed the similarities between the theory of codes and the theory of Euclidean lattices. It is interesting to compare the two theories since they share a common problem, viz. the sphere packing problem. In the theory of codes one would like to find a code over Fp, i.e. a subspace of Fpn, such that non-intersecting spheres with respect to a given metric, centered at the code vectors, pack Fpndensely.

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The Expanding Space Method in Sphere Packing Problem

Advances in Intelligent Systems and Computing - Lecture Notes in Computational Intelligence and Decision Making ◽

10.1007/978-3-030-54215-3_10 ◽

2020 ◽

pp. 151-163

Author(s):

Sergiy Yakovlev

Keyword(s):

Sphere Packing ◽

Packing Problem ◽

Space Method

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The Sphere Packing Problem

Pedagogy and Content in Middle and High School Mathematics ◽

10.1007/978-94-6351-137-7_23 ◽

2017 ◽

pp. 93-93

Author(s):

G. Donald Allen

Keyword(s):

Sphere Packing ◽

Packing Problem

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Embryonic cleavage modeling as a computational approach to sphere packing problem

Journal of Theoretical Biology ◽

10.1016/j.jtbi.2006.09.032 ◽

2007 ◽

Vol 245 (1) ◽

pp. 77-82 ◽

Cited By ~ 4

Author(s):

Luca Zammataro ◽

Guido Serini ◽

Todd Rowland ◽

Federico Bussolino

Keyword(s):

Sphere Packing ◽

Packing Problem ◽

Computational Approach ◽

Embryonic Cleavage

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Monte Carlo study of the sphere packing problem

Physica A Statistical Mechanics and its Applications ◽

10.1016/s0378-4371(02)01798-3 ◽

2003 ◽

Vol 321 (1-2) ◽

pp. 359-363 ◽

Cited By ~ 7

Author(s):

S.P Li ◽

Ka-Lok Ng

Keyword(s):

Monte Carlo ◽

Sphere Packing ◽

Packing Problem ◽

Monte Carlo Study

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Fourier uniqueness in even dimensions

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.2023227118 ◽

2021 ◽

Vol 118 (15) ◽

pp. e2023227118

Author(s):

Andrew Bakan ◽

Haakan Hedenmalm ◽

Alfonso Montes-Rodríguez ◽

Danylo Radchenko ◽

Maryna Viazovska

Keyword(s):

Modular Forms ◽

Short Note ◽

Sphere Packing ◽

Packing Problem ◽

Uniqueness Results ◽

Alternative Approach ◽

Klein Gordon ◽

Gauss Type ◽

Key Dimensions ◽

Uniqueness Theory

In recent work, methods from the theory of modular forms were used to obtain Fourier uniqueness results in several key dimensions (d=1,8,24), in which a function could be uniquely reconstructed from the values of it and its Fourier transform on a discrete set, with the striking application of resolving the sphere packing problem in dimensions d=8 and d=24. In this short note, we present an alternative approach to such results, viable in even dimensions, based instead on the uniqueness theory for the Klein–Gordon equation. Since the existing method for the Klein–Gordon uniqueness theory is based on the study of iterations of Gauss-type maps, this suggests a connection between the latter and methods involving modular forms. The derivation of Fourier uniqueness from the Klein–Gordon theory supplies conditions on the given test function for Fourier interpolation, which are hoped to be optimal or close to optimal.

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