The $D_4$ Root System Is Not Universally Optimal

Henry Cohn; John H. Conway; Noam D. Elkies; Abhinav Kumar

doi:em/1204928532

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Home > Journals > Experiment. Math. > Volume 16 > Issue 3 > Article

2007 The

D_{4}

Root System Is Not Universally Optimal

Henry Cohn, John H. Conway, Noam D. Elkies, Abhinav Kumar

Experiment. Math. 16(3): 313-320 (2007).

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Abstract

We prove that the $D_{4}$ root system (equivalently, the set of vertices of the regular 24-cell) is not a universally optimal spherical code. We further conjecture that there is no universally optimal spherical code of 24 points in $S^{3}$ , based on numerical computations suggesting that every 5-design consisting of 24 points in $S^{3}$ is in a 3-parameter family (which we describe explicitly, based on a construction due to Sali) of deformations of the $D_{4}$ root system.

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Henry Cohn. John H. Conway. Noam D. Elkies. Abhinav Kumar. "The $D_{4}$ Root System Is Not Universally Optimal." Experiment. Math. 16 (3) 313 - 320, 2007.