Indicators

[Leonard Soicher]

On Thu, Oct 02, 2003 at 02:44:01PM +0100, Peter Cameron wrote:
[...]

> The other reason I don't like this is that "affine" is so well-established
> and has such a body of theory that I would be reluctant to change the
> usage. On the other side, designs with block intersections of size
> \alpha or \beta are well studied: they are called "quasi-symmetric" (at
> least if they are 2-designs, and I think the generalisation here is 
> unexceptionable), and for example Shrikhande and Sane have a book about
> them.
> 

Let me remind everyone (including myself) of a number of points:

(1) We can always add indicators at a later date, but it is difficult to 
take one away. Thus, if in doubt about an indicator (or any other element)
we should leave it out (for now).

(2) We put a lot of thought into our function_on_... definitions
to allow for all kinds of useful information to be displayed in
an appropriate and compact way.  In particular, according to our
implementation policies, for pairwise block concurrences (intersection
sizes), we store the intersection sizes together with the number of
pairs having each intersection size. This nicely handles JP's notion of affine,
the notion of "quasi-symmetric", and future generalizations.

I thus vote no to an "affine" indicator (at least for the first release
of the ext-rep).  

By the way, I am opposed to an indicator consisting of more than a flag and
(possibly) a useful additional attribute containing a single simple value
(such as a non-negative integer).

[...]

> 2. Partially balanced
> ---------------------
> For general block designs, there are (at least) three different kinds of
> balance: pairwise balance, variance balance, and efficiency balance. If
> the block size is constant these all reduce to the same thing.
> 
> May something similar be true for partial balance? Thus, "pairwise partial
> balance" means that the concurrence matrix is in the Bose-Mesner algebra
> of an association scheme; "variance partial balance" means that the 
> information matrix is (do I have this right JP?) and there may be a kind of 
> "efficiency partial balance" as well.
> 
> My proposal is that we do not in ignorance assume that only one of these
> three concepts is important. We could easily have three indicators as
> outlined above. 

Until we do further research on this interesting complication, I propose
we remove the partially_balanced indicator from the first release of the 
ext-rep.

[...]

Leonard