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原文摘要

We give explicit kissing configurations in R^d for every d = 49, ..., 61. The lower bound is improved over the best previously published value in all thirteen dimensions, by margins from 232 to 159,220 points and by 506,461 points in total. The configurations are obtained from the antipodal block lift of Yang et al. applied to the 52,416,000-point minimal shell of the extremal even unimodular lattice P_48p, using a class of 7,193 minimal lines and, in each dimension, a kissing configuration of R^k (k <= 15) with the largest available number of antipodal pairs. Two ingredients are new. The published class of 7,077 lines is grown to 7,193 by local search in three stages, as a sequence of exchanges between candidate lines and class lines; and the point count, which equals 52,416,000 + 2S + 2t with t the number of antipodal pairs and S the size of the union of t isometric images of the class, is maximised as a maximum-coverage problem by greedy search with iterated local search over pools of 4,000 automorphic images, reaching 92-100% of the trivial bound 7,193t. All results are verified in exact integer arithmetic, and the certificates are distributed with the code. Code and data: https://github.com/dimkadimon/KissingNumbers