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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
On the number of nonequivalent propelinear extended perfect codes
The paper proves that there exist an exponential number of nonequivalent
propelinear extended perfect binary codes of length growing to infinity.
Specifically, it is proved that all transitive extended perfect binary codes
found by Potapov are propelinear. All such codes have small rank, which is one
more than the rank of the extended Hamming code of the same length. We
investigate the properties of these codes and show that any of them has a
normalized propelinear representation
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