OBIBDs constructed from rings
Notation for lists. If H denotes a list of elements,
yH denotes the list obtained by premultiplying the
elements of H by y. ΔH is defined to be the
set of differences between pairs {hi -
hj: hi,hj
∈ H}. If G,H are two lists, then GH is
defined to be the list of all products
{gihj:
gi∈G, hj
∈H}. A list comprising n copies of H is
denoted by n(H). If H1,
H2,..,Hn are lists, then
∑Hi denotes the concatenation of these
lists.
- OBIBD (Pergola, even) construction based on rings
- Informal Construction.The construction generalizes a
BIBD construction due to
Furino,
and the notation used here is
similar. There are clear analogies with the OBIBD constructions
of
Morgan & Uddin.
Suppose a ring R with |R| =
v, that the units of R be given by U(R),
that U(R) has a subgroup B of composite order, say
hf, and that the differences between the elements of
B, ΔB, are also members of U(R). The
design is based upon a subgroup C of order f of
B. This will generate the blocks for the first set of
treatments, and the designs for the other sets of treatments are
generated from the cosets of C in B. Let a typical
coset of C in B be written as xC,
where x belongs to B. Let S be a set of
distinct representatives for C in R, and let
T⊂(S∩U(R)), where |T| =
e. Let F = ∪tiC where
ti∈T. If ΔF
⊂U(R), then a set of initial blocks (a difference
family) for the first set of treatments is given by Φ =
{sF : s∈S}. A difference family
for a further set of treatments is given by xΦ =
{sxF : s∈S} where x is a
coset representative for C in B.
- The proof that the various difference families generate BIBDs
is as in Furino. It remains to show that the relatiuonship
between the difference families is as required. Let
Δ(C,xC) =
{ci-xcj}, where
ci, cj range over the
elements of C, and define
Δ(F,xF) and
Δ(Φ,xΦ) accordingly.
- Then ΔF =
∑∑(xtic-tj)C,
and Δ(Φ,xΦ)=
∑∑∑s(xtic-tj)
C =
∑∑(xtic-tj)∑s
C =
∑∑(xtic-tj){R\{0}}.
This gives e2f occurrences of the
non-zero elements of R, where the block size is
ef.
- Now define Δ*(C,xC) =
{ci-xci :
ci∈C}, and define Δ*(F,xF)
and Δ*(Φ,xΦ) accordingly. Then
ΔF= ∑(ti(1-x)C),
and Δ(Φ,xΦ)=
∑∑sti(1-x)C =
∑ti∑s(1-x)C =
∑ti{R\{0}}. And this gives e
occurrences of the non-zero elements of R.
Some examples are given here. This discussion is
continued here. A table
giving suitable inputs for solutions, with v up to 500, is
given here.