Indicators

Indicators are boolean variables which record certain properties which a block design may have. We have included the following indicators:

repeated_blocks

True if the same set occurs more than once in the list of blocks.

resolvable

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.

affine_resolvable

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 $\mu$ of points. If the design is affine resolvable then we optionally give this constant $\mu$ (unless the design consists of a single parallel class, in which case $\mu$ is not defined).

equireplicate

True if each point lies in a fixed number $r$ of blocks. If so, then we also optionally give the replication number $r$.

constant_blocksize

True if each block contains a fixed number $k$ of points. If so, then we optionally also give the block size $k$.

t_design

True if the block design is a t-design for some $t > 1$. This means that the design has constant block size and that any $t$ points are contained in a positive constant number $\lambda$ of blocks. If so, then we optionally give the maximum value of $t$ for which this holds.

connected

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.

pairwise_balanced

True if $v>1$ and the number of blocks containing two distinct points is a positive constant $\lambda$. If so, then we optionally give this $\lambda$.

variance_balanced

True if $v>1$ and the intra-block information matrix has $v-1$ identical, nonzero eigenvalues. Equivalently, the $v-1$ canonical variances are all equal (and finite). For definitions of terms used here, see section 7.6 on Statistical Properties.

efficiency_balanced

True if $v>1$ and the $v-1$ 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.

cyclic

True if the design has an automorphism which permutes all the points in a single cycle.

one_rotational

True if the design has an automorphism which fixes one point and permutes the other $v-1$ 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>