[seqfan] Groups from Grassmannian packings: a question
Simone Severini
simoseve at gmail.com
Sat Nov 27 16:38:56 CET 2010
Dear SeqFans,
Ashkhmin and Gopalan (see reference below) defined a family of complex
vector quantization codebooks based on Grassmannian packings and Reed-Muller
codes, binary and over Z_4. (A. Ashikhmin and R. Gopalan, Grassmannian
packings for efficient quantization in MIMO broadcast systems, In IEEE
International Symposium on Information Theory, pages 1811--1815, 2007.)
The orthogonality graph of a set of complex vectors is the graph whose
vertices are the vectors, two vertices being adjacent if the vectors are
orthogonal.
I am interested in the orthogonality graphs G_m whose vertices are the
codewords of the Ashkhmin-Gopalan codes.
It is immediate to observe that the graph G_m are k-regular and on n
vertices, where
k = \prod_{j=0}^{m-1}( 2^{m-j}+1)
(which is 3,15,135,...)
n = 2^{m}\prod_{k=0}^{m-1}( 2^{m-k}+1)
(which is 6,60, 1080,...)
The authomorphism group of G_2 (15-regular on 60-vertices) has order 23040
(computed with the SAGE web interface).
I would like to know the following:
Is each G_m a Cayley graph? If this is the case, what are the associated
groups?
(It should be the case, but I am not sure about the groups.)
I have constructed the adjacency matrices for small m.
Thanks a lot for your help.
Best regards,
Simone
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Simone Severini
http://www.homepages.ucl.ac.uk/~ucapsse<http://www.homepages.ucl.ac.uk/%7Eucapsse>
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