[GAP Forum] Parker loop

[Stephen Linton]

You can work with the Parker loop efficiently without storing its entire multiplication table.
I've forgotten the details now but there is a function \Theta from the code to the cocode and 
the sign adjustment in the multiplication turns out to be -1^|\Theta(c1)  /\  c2| where c1 and c2 are codewords, considered as sets.

	Steve
On 20 Dec 2011, at 15:38, Marek Mitros wrote:

> 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
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>>> Forum mailing list
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>> 
>> 
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