#16866: Radical difference families
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Reporter: vdelecroix | Owner:
Type: enhancement | Status: needs_info
Priority: major | Milestone: sage-6.4
Component: combinatorial | Resolution:
designs | Merged in:
Keywords: | Reviewers:
Authors: Vincent Delecroix | Work issues:
Report Upstream: N/A | Commit:
Branch: | 721af75ec2b2c6ca904f2feaf86e75e054cb089d
u/vdelecroix/16866 | Stopgaps:
Dependencies: #16863 |
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Comment (by vdelecroix):
Replying to [comment:24 ncohen]:
> > No. I want that all differences belong to different coset.
>
> Why can you make that assumption ?
Because you want a difference family at the end! Let me recall that B =
\Delta H^2mt^ = A H^mt^ is your basic block (see below for explanation).
If two class of A belong to the same set modulo H^mt^ them it means that
in the product A H^mt^ you have zero. Which contradict the definition of
difference family which needs to cover K \ {0}.
> > This comes from computing what is Delta {1, r, r^2^, ..., r^k-1^}
(where r is a k-th root of unity). If you do that you will see that it is
the same as {+1, -1} A H^mt^. In the case of k even you want to compute
Delta {0, 1, r, ..., r^k-2^} (where r is a (k-1)-th root of unity).
>
> I have absolutely no intuition of what H^mt^ represents.
H is the set of invertible elements in the finite field (it has cardinal
q-1 and form a group under multiplication. H^j^ is the set of j-th power
in H.
In our context, H^mt^ is the set of (2k)-th root of unity or 2(k-1)-th
root of unity depending on the parity of k. It is also {+1, -1} H^2mt^
where H^2mt^ is the set of k-th root of unity or (k-1)-th root of unity
(depending on the parity.
> > > And so far I do not understand how your code works.
> >
> > Good point
>
> Do you think that the tiling problem that you solve is equivalent, by
the previous remarks, to the non-reduced problem ? If so, that would make
it easier for me to understand.
Yes. If you solve the big one, you solve the small one (and conversly).
Vincent
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Ticket URL: <http://trac.sagemath.org/ticket/16866#comment:25>
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