Indicators are boolean variables which record certain properties which a block design may have. We have included the following indicators:
True if the same set occurs more than once in the list of blocks.
True if the design has a resolution, which is a partition of the blocks into subsets called parallel classes or resolution classes, each of which forms a partition of the point set.
True if the design is affine resolvable, which means that the design is resolvable and any two blocks not in the same parallel class of a resolution meet in a constant number of points. If the design is affine resolvable then we optionally give this constant (unless the design consists of a single parallel class, in which case is not defined).
True if each point lies in a fixed number of blocks. If so, then we also optionally give the replication number .
True if each block contains a fixed number of points. If so, then we optionally also give the block size .
True if the block design is a t-design for some . This means that the design has constant block size and that any points are contained in a positive constant number of blocks. If so, then we optionally give the maximum value of for which this holds.
True if the incidence graph of the block design is a connected graph. (The incidence graph or Levi graph of a block design is the bipartite graph whose vertices are the points and blocks of the design, a point and block being adjacent if the point is contained in the block.) We optionally give the number of connected components of the incidence graph.
True if and the number of blocks containing two distinct points is a positive constant . If so, then we optionally give this .
True if and the intra-block information matrix has identical, nonzero eigenvalues. Equivalently, the canonical variances are all equal (and finite). For definitions of terms used here, see section 7.6 on Statistical Properties.
True if and the statistical canonical efficiency factors are identical and nonzero. For equireplicate designs, this is equivalent to variance_balanced, but not genenerally otherwise. Also see the section 7.6 Statistical Properties.
True if the design has an automorphism which permutes all the points in a single cycle.
True if the design has an automorphism which fixes one point and permutes the other points in a single cycle.
In the last two cases, an automorphism with the stated properties can be found under cycle_type_representatives, described in section 7.4 on Automorphisms.
The several different sorts of balance are explained in the http://designtheory.org/library/encycEncyclopaedia. For a (binary) design with constant block size, variance balance reduces to pairwise balance. For a equireplicate (binary) design with constant block size, efficiency balance reduces to pairwise balance.
The indicators for our example are:
<indicators> <repeated_blocks flag="false"> </repeated_blocks> <resolvable flag="false"> </resolvable> <affine_resolvable flag="false"> </affine_resolvable> <equireplicate flag="true" r="3"> </equireplicate> <constant_blocksize flag="true" k="3"> </constant_blocksize> <t_design flag="true" maximum_t="2"> </t_design> <connected flag="true" no_components="1"> </connected> <pairwise_balanced flag="true" lambda="1"> </pairwise_balanced> <variance_balanced flag="true"> </variance_balanced> <efficiency_balanced flag="true"> </efficiency_balanced> <cyclic flag="true"> </cyclic> <one_rotational flag="false"> </one_rotational> </indicators>