Hi All,
For some time I struggle to obtain Golay code from the Mathieu group and
opposite - find Mathieu group which preserve given Golay code. I found
solution in one way which I present below. It is quick. Does anybody know
how to obtain Mathieu Group M24 generators for given Golay code ?
Regards,
Marek Mitros
#--------------------------------------------------------------------------------------------------------
m:=MathieuGroup(24);
stab:=Stabilizer(m, [1,2,3,4,5], OnTuples);
n0:=First([6..24], x->Size(Orbit(stab,x))=3); # find 3-element orbit
# now we have first octad
o1:=Concatenation([1..5], Orbit(stab, n0));
rs:=GlobalRandomSource;
rand:=function(n) return Random(rs, [1..n]); end;;
gen:=GeneratorsOfGroup(m); # 3 generators of order 23,5,2
octads:=[o1]; count:=0;
while Size(octads)<759 and count <20000 do
# generate random element
rl:=List([1..10], n->rand(3));
g:=Product(gen{rl});
# find image of the octad
im:=OnSets(o1, g);
if not (im in octads) then
Add(octads, im);
fi;
count:=count+1;
if count mod 1000 = 0 then Print(Size(octads), " octads found; counter=",
count,"\n"); fi;
od;
# now we have ready 759 octads. Verify result whether intersections of
octads are 0,2,4
Set(List(octads, x->Size(Intersection(octads[1], x))))= [0,2,4,8];
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