TY - JOUR

T1 - Equiangular lines and the Lemmens–Seidel conjecture

AU - Lin, Yen Chi Roger

AU - Yu, Wei Hsuan

N1 - Funding Information:
The authors thank Eiichi Bannai and Alexey Glazyrin for their helpful discussions on this work. This material is based upon work supported by the National Science Foundation, United States under Grant No. DMS-1439786 while the second author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Point configurations in Geometry, Physics and Computer Science Program. Part of this work was done when the second author visited National Center for Theoretical Sciences (NCTS), Taiwan, in the summer of 2018. The authors are grateful to the support of NCTS. Finally, the authors would like to thank anonymous referees who offer useful comments on improvement of Theorem 5.3 and give clues to the remark following Theorem 5.3.
Funding Information:
The authors thank Eiichi Bannai and Alexey Glazyrin for their helpful discussions on this work. This material is based upon work supported by the National Science Foundation, United States under Grant No. DMS-1439786 while the second author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Point configurations in Geometry, Physics and Computer Science Program. Part of this work was done when the second author visited National Center for Theoretical Sciences (NCTS), Taiwan, in the summer of 2018. The authors are grateful to the support of NCTS. Finally, the authors would like to thank anonymous referees who offer useful comments on improvement of Theorem 5.3 and give clues to the remark following Theorem 5.3 .
Publisher Copyright:
© 2019 Elsevier B.V.

PY - 2020/2

Y1 - 2020/2

N2 - In this paper, claims by Lemmens and Seidel in 1973 about equiangular sets of lines with angle 1∕5 are proved by carefully analyzing pillar decomposition, with the aid of the uniqueness of two-graphs on 276 vertices. The Neumann Theorem is generalized in the sense that if there are more than 2r−2 equiangular lines in Rr, then the angle is quite restricted. Together with techniques on finding saturated equiangular sets, we determine the maximum size of equiangular sets “exactly” in an r-dimensional Euclidean space for r=8, 9, and 10.

AB - In this paper, claims by Lemmens and Seidel in 1973 about equiangular sets of lines with angle 1∕5 are proved by carefully analyzing pillar decomposition, with the aid of the uniqueness of two-graphs on 276 vertices. The Neumann Theorem is generalized in the sense that if there are more than 2r−2 equiangular lines in Rr, then the angle is quite restricted. Together with techniques on finding saturated equiangular sets, we determine the maximum size of equiangular sets “exactly” in an r-dimensional Euclidean space for r=8, 9, and 10.

KW - Equiangular set

KW - Lemmens–Seidel

KW - Pillar methods

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U2 - 10.1016/j.disc.2019.111667

DO - 10.1016/j.disc.2019.111667

M3 - Article

AN - SCOPUS:85072609684

VL - 343

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 2

M1 - 111667

ER -