[GAP Forum] Parker loop

[Marek Mitros]

Thank you for answers from Steve and Gabor.

I wonder following thing - maybe any of you can comment on it. Let Ld be
left multiplication in Parker loop - it is operator in R^4096. From Conway
paper I assume that L_d for d in Parker loop and 2^12 diagonal
automorphisms x_delta (defined in chapter 4 of Conway) generate
extraspecial group 2^25. I assume following
L_o for octad o square to 1;
L_d for dodecad d square to -1;
if a,b are two octads intersecting in 4 points then L_a and L_b commute;
if a is octad and b is dodecad then probably L_a and L_b anticommute...

In order to prove that group defined above is extraspecial 2^(1+24) we
should find convenient basis of Golay code {a} so then L_a and some 12
diagonal automorphisms generate Clifford C(24) monomials which is
extraspecial 2^25. C(24) = M(4096). I found one nice base in Tsu paper from
1998. Is there exist basis of Golay code with octads only ? There should
be, because I just tested in GAP that dimension of vector space generated
by 759 octads is 12 over GF(2).

Regarding code from Gabor. I had out of memory error as well when trying to
define Parker loop. I was able to store table 8192 x 8192 on the disk - it
is 394 MB size (!). I wonder what is the mapping of indexes 1..8192 with
+-d for d in Golay Code. There should be one row with 1..8192 numbers but
it is not the first one. I have sent email to Gabor but no answer so far,
so I post it here.

Regards,
Marek



On Fri, Dec 2, 2011 at 12:14 PM, Stephen Linton <sal at cs.st-andrews.ac.uk>wrote:

> I started trying to automatise this construction some years ago. I recall
> that I got as far as constructing Fi24 in characteristic zero
> by these methods. I don't know if I still have any of the code I wrote,
> let alone whether it works with current GAP.
> I'll have a look.
>
>        Steve
>
> On 2 Dec 2011, at 10:52, Marek Mitros wrote:
>
> > Hi All,
> >
> > I am reading Conway "simple construction of monster" and I wonder whether
> > anybody has Parker loop defined for GAP. I would like to play around with
> > it to understand more how multiplication there looks like.
> > For example when I have two octads o1, o2 intersecting in four points
> then
> > let o3 be XOR(o1,o2). Then I assume that in Parker loop o1.o2=o3,
> o2.o3=o1,
> > etc and all these products commutes.
> > When I have octads o1,o2 intersecting in two points then d1=XOR(o1,o2) is
> > dodecad. In such case I do not know whether o1.o2=d1 or o1.o2=-d1
> (minus).
> > In such case product o1.o2 anticommute.
> >
> > The next question I have is how to generate extraspecial group of size
> 2^25
> > called Q_x1 in the paper. It is generated by elements x_d and x_delta.
> >
> > Regards,
> > Marek
> > _______________________________________________
> > Forum mailing list
> > Forum at mail.gap-system.org
> > http://mail.gap-system.org/mailman/listinfo/forum
>
>