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We prove that the Cohn–Elkies linear programming bound for sphere packing is not sharp in dimension 6. The proof uses duality and optimization over a space of modular forms, generalizing a construction of Cohn–Triantafillou Math. Comp. 91 (2021), pp. 491–508 to the case of odd weight and non-trivial character.
Courcy-Ireland et al. (Wed,) studied this question.
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