6,092 research outputs found
Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups
In this paper, we present a method to obtain regular (or equitable)
partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of
permutation groups on letters. We prove that every partition of the number
gives rise to a regular partition of the Cayley graph. By using
representation theory, we also obtain the complete spectra and the eigenspaces
of the corresponding quotient (di)graphs. More precisely, we provide a method
to find all the eigenvalues and eigenvectors of such (di)graphs, based on their
irreducible representations. As examples, we apply this method to the pancake
graphs and to a recent known family of mixed graphs
(having edges with and without direction). As a byproduct, the existence of
perfect codes in allows us to give a lower bound for the multiplicity of
its eigenvalue
The Perfect Binary One-Error-Correcting Codes of Length 15: Part II--Properties
A complete classification of the perfect binary one-error-correcting codes of
length 15 as well as their extensions of length 16 was recently carried out in
[P. R. J. \"Osterg{\aa}rd and O. Pottonen, "The perfect binary
one-error-correcting codes of length 15: Part I--Classification," IEEE Trans.
Inform. Theory vol. 55, pp. 4657--4660, 2009]. In the current accompanying
work, the classified codes are studied in great detail, and their main
properties are tabulated. The results include the fact that 33 of the 80
Steiner triple systems of order 15 occur in such codes. Further understanding
is gained on full-rank codes via switching, as it turns out that all but two
full-rank codes can be obtained through a series of such transformations from
the Hamming code. Other topics studied include (non)systematic codes, embedded
one-error-correcting codes, and defining sets of codes. A classification of
certain mixed perfect codes is also obtained.Comment: v2: fixed two errors (extension of nonsystematic codes, table of
coordinates fixed by symmetries of codes), added and extended many other
result
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