#20968: more Hadamard matrices with constant diagonal
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Reporter: dimpase | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-7.3
Component: graph theory | Resolution:
Keywords: | Merged in:
Authors: Dima Pasechnik | Reviewers: Vincent Delecroix
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/dimpase/hadaconst | 7509fc0d47d720064a6a36dad342adee033a9222
Dependencies: | Stopgaps:
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Comment (by dimpase):
Replying to [comment:25 dimpase]:
> Replying to [comment:24 vdelecroix]:
> > In `szekeres_difference_set_pair` it would be better to actually check
that the output is what you want. In other words check that `A` and `B`
are difference sets and that there are complementary.
>
> it is actually even more than that; A has to be skew-symmetric and B has
to be
> symmetric (in terms of their "typeI matrices"). But this, as well as
the above, is
> tested indirectly, by building the corresponding Hadamard matrix.
>
These sets are not difference sets in the
[https://en.wikipedia.org/wiki/Difference_set classical sense]. Indeed,
{{{
sage: szekeres_difference_set_pair(2)
([1, 3, 4, 5, 9], [4, 5], [3, 4])
}}}
That is you have v=5, k=2, and the standard equation k^2^-k=(v-1)l on the
parameters (v,k,l) of a difference set implies l=1/2, which just cannot
happen.
What we have here is a difference family, with s=2, so you have
s(k^2^-k)=(v-1)l, ie
[https://en.wikipedia.org/wiki/Difference_set#Generalisations the
parameters of the family] are (v,k,l,s)=(2m+1,m,1,2). Note that in
`designs.difference_family` the parameter s is not explicit (but can be
computed as (v-1)l/(k^2^-k)).
--
Ticket URL: <https://trac.sagemath.org/ticket/20968#comment:26>
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