Dual linear programming bounds for sphere packing via modular forms

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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Bounds on Subspace Codes Based on Orthogonal Space Over Finite Fields of Characteristic 2

International Journal of Foundations of Computer Science ◽

10.1142/s0129054119500199 ◽

2019 ◽

Vol 30 (05) ◽

pp. 735-757

Author(s):

Gang Wang ◽

Min-Yao Niu ◽

Fang-Wei Fu

Keyword(s):

Linear Programming ◽

Positive Integer ◽

Finite Fields ◽

Sphere Packing ◽

Singular Subspace ◽

Subspace Codes ◽

Orthogonal Space ◽

Johnson Bound ◽

Characteristic 2 ◽

Subspace Distance

In this paper, the Sphere-packing bound, Wang-Xing-Safavi-Naini bound, Johnson bound and Gilbert-Varshamov bound on the subspace code of length [Formula: see text], size [Formula: see text], minimum subspace distance [Formula: see text] based on [Formula: see text]-dimensional totally singular subspace in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over finite fields [Formula: see text] of characteristic 2, denoted by [Formula: see text], are presented, where [Formula: see text] is a positive integer, [Formula: see text], [Formula: see text], [Formula: see text]. Then, we prove that [Formula: see text] codes attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in [Formula: see text], where [Formula: see text] denotes the collection of all the [Formula: see text]-dimensional totally singular subspaces in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over [Formula: see text] of characteristic 2. Finally, Gilbert-Varshamov bound and linear programming bound on the subspace code [Formula: see text] in [Formula: see text] are provided, where [Formula: see text] denotes the collection of all the totally singular subspaces in the [Formula: see text]-dimensional orthogonal space [Formula: see text] over [Formula: see text] of characteristic 2.

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High-dimensional sphere packing and the modular bootstrap

Journal of High Energy Physics ◽

10.1007/jhep12(2020)066 ◽

2020 ◽

Vol 2020 (12) ◽

Author(s):

Nima Afkhami-Jeddi ◽

Henry Cohn ◽

Thomas Hartman ◽

David de Laat ◽

Amirhossein Tajdini

Keyword(s):

Linear Programming ◽

Numerical Study ◽

Sphere Packing ◽

High Dimensional ◽

Dimensional Sphere ◽

Conformal Field Theories ◽

Spherical Codes ◽

Conformal Field ◽

Detailed Picture ◽

Rule Out

Abstract We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra U(1)c× U(1)c, or equivalently the linear programming bound for sphere packing in 2c dimensions. We give a more detailed picture of the behavior for finite c than was previously available, and we extrapolate as c → ∞. Our extrapolation indicates an exponential improvement for sphere packing density bounds in high dimen- sions. Furthermore, we study when these bounds can be tight. Besides the known cases c = 1/2, 4, and 12 and the conjectured case c = 1, our calculations numerically rule out sharp bounds for all other c < 90, by combining the modular bootstrap with linear programming bounds for spherical codes.

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