[seqfan] Groups from Grassmannian packings: a question

[Simone Severini]

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
---------------------------------------------------------------
Simone Severini
http://www.homepages.ucl.ac.uk/~ucapsse<http://www.homepages.ucl.ac.uk/%7Eucapsse>
---------------------------------------------------------------