Indicators

[Leonard Soicher]

Dear All, 

This email concerns the specific merits of the "affine" indicator that JP
has suggested, whatever we may choose to call it. Recall JP's suggestion:

 
> 2.  Add the indicator "affine":  true if and only if any two
> nonparallel blocks meet in the same number \mu of points.
> Analogous to the t-wise balance indicator, include the value
> of \mu.
> 
> Reasoning:  aside from intrinsic interest, this allows other
> important properties to be deduced from the set of
> indicators.  For example,
> affine+resolvable+constant_blocksize is equivalent to an
> orthogonal array of strength 2 and index \mu.  If 2-wise
> balance also holds and \mu=1, then the design is equivalent
> to a complete sets of MOLS, etc, etc.
> 

Following from the comments of Peter Cameron, this does not appear to
be the case.  For example, how is a resolvable 2-(15,3,1) equivalent to
a complete set of MOLS? So it seems that the "affine-like" indicator,
if we have one, should be true iff the block design is resolvable and
any two blocks in different parallel classes of a resolution meet in a
constant number \mu of points.  I would support such an "affine-like"
indicator.  Would others?

Leonard.