Frames over finite fields: equiangular lines in orthogonal geometry
We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogon...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
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2022
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| Online Access: | https://hdl.handle.net/10356/162564 |
| _version_ | 1811697456452206592 |
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| author | Greaves, Gary Royden Watson Iverson, Joseph W. Jasper, John Mixon, Dustin G. |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Greaves, Gary Royden Watson Iverson, Joseph W. Jasper, John Mixon, Dustin G. |
| author_sort | Greaves, Gary Royden Watson |
| collection | NTU |
| description | We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but 18 known SRG parameters on v≤1300 vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzon's bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzon's bound cannot be attained in a finite orthogonal geometry of dimension 5. |
| first_indexed | 2024-10-01T07:55:33Z |
| format | Journal Article |
| id | ntu-10356/162564 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T07:55:33Z |
| publishDate | 2022 |
| record_format | dspace |
| spelling | ntu-10356/1625642022-10-31T01:56:37Z Frames over finite fields: equiangular lines in orthogonal geometry Greaves, Gary Royden Watson Iverson, Joseph W. Jasper, John Mixon, Dustin G. School of Physical and Mathematical Sciences Science::Mathematics Equiangular Lines Equiangular Tight Frames We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but 18 known SRG parameters on v≤1300 vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzon's bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzon's bound cannot be attained in a finite orthogonal geometry of dimension 5. Ministry of Education (MOE) GRWG was partially supported by the Singapore Ministry of Education Academic Research Fund (Tier 1); grant numbers: RG29/18 and RG21/20. JJ was supported by NSF DMS 1830066. DGM was partially supported by AFOSR FA9550-18-1-0107 and NSF DMS 1829955. 2022-10-31T01:56:36Z 2022-10-31T01:56:36Z 2022 Journal Article Greaves, G. R. W., Iverson, J. W., Jasper, J. & Mixon, D. G. (2022). Frames over finite fields: equiangular lines in orthogonal geometry. Linear Algebra and Its Applications, 639, 50-80. https://dx.doi.org/10.1016/j.laa.2021.11.024 0024-3795 https://hdl.handle.net/10356/162564 10.1016/j.laa.2021.11.024 2-s2.0-85122642661 639 50 80 en RG29/18 RG21/20 Linear Algebra and Its Applications © 2022 Elsevier Inc. All rights reserved. |
| spellingShingle | Science::Mathematics Equiangular Lines Equiangular Tight Frames Greaves, Gary Royden Watson Iverson, Joseph W. Jasper, John Mixon, Dustin G. Frames over finite fields: equiangular lines in orthogonal geometry |
| title | Frames over finite fields: equiangular lines in orthogonal geometry |
| title_full | Frames over finite fields: equiangular lines in orthogonal geometry |
| title_fullStr | Frames over finite fields: equiangular lines in orthogonal geometry |
| title_full_unstemmed | Frames over finite fields: equiangular lines in orthogonal geometry |
| title_short | Frames over finite fields: equiangular lines in orthogonal geometry |
| title_sort | frames over finite fields equiangular lines in orthogonal geometry |
| topic | Science::Mathematics Equiangular Lines Equiangular Tight Frames |
| url | https://hdl.handle.net/10356/162564 |
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