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Frames, Group Codes, and Subgroups of (Z/pZ)×

Thill, Matthew and Hassibi, Babak (2012) Frames, Group Codes, and Subgroups of (Z/pZ)×. In: 2012 50th Annual Allerton Conference on Communication, Control, and Computing. IEEE , Piscataway, NJ, pp. 1182-1189. ISBN 978-1-4673-4537-8. https://resolver.caltech.edu/CaltechAUTHORS:20130730-143016951

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Abstract

The problem of designing low coherence matrices and low-correlation frames arises in a variety of fields, including compressed sensing, MIMO communications and quantum measurements. The challenge is that one must control the (^n_2) pairwise inner products of the columns of the matrix. In this paper, we follow the group code approach of David Slepian [1], which constructs frames using unitary group representations and which in general reduces the number of distinct inner products to n-1. When n is a prime p, we present a carefully chosen representation which reduces the number of distinct inner products further to ^(n-1)/_m, where m is the number of rows in the matrix. The resulting frames have superior performance to many earlier frame constructions and, in some cases, yield frames with optimally low coherence. We further expand a connection between frames and difference sets noted first in [2] to find bounds on the coherence when ^(n-1)/_m = 2 and 3.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1109/Allerton.2012.6483352DOIArticle
ORCID:
AuthorORCID
Thill, Matthew0000-0003-0885-6260
Additional Information:© 2012 IEEE. This work was supported in part by the National Science Foundation under grants CCF-0729203, CNS-0932428 and CCF-1018927, by the Office of Naval Research under the MURI grant N00014-08-1-0747, and by Caltech's Lee Center for Advanced Networking. The first author was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program.
Funders:
Funding AgencyGrant Number
NSFCCF-0729203
NSFCNS-0932428
NSFCCF-1018927
Office of Naval Research (ONR)N00014-08-1-0747
Caltech Lee Center for Advanced NetworkingUNSPECIFIED
National Defense Science and Engineering Graduate (NDSEG) FellowshipUNSPECIFIED
Subject Keywords:Matrix coherence, frame, Grassmannian frame, unit norm tight frame, compressed sensing, group, unitary system
DOI:10.1109/Allerton.2012.6483352
Record Number:CaltechAUTHORS:20130730-143016951
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20130730-143016951
Official Citation:Thill, M.; Hassibi, B., "Frames, group codes, and subgroups of (Z/pZ)×," Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on , vol., no., pp.1182,1189, 1-5 Oct. 2012 doi: 10.1109/Allerton.2012.6483352
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:39661
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:30 Jul 2013 22:08
Last Modified:09 Nov 2021 23:46

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