The proof relies on some constructions which can be found in the
following:

G.R.\Chaudry, M.\ Greig and J.\ Seberry,
On the (v,5,$\lambda$)-Family of Bhaskar Rao Designs,
(1998), (submitted).

In particular we term a signed GDD$_{\lambda}(k,$ "group type" $)$
whose incidence matrix satisfies $NN^T=rI$ as a BRGDD$_{\lambda}(k,$
"group type" $)$; we can use these BRGDDs as either ingredients or
master designs (or both) in a variant of Wilson's fundamental
construction (see CRC[III.2.5] for original version, and~\cite{cgs}
for the variant).

Use a BRGDD$_2(4,2^4)$ as the master with weight 3 and ingredient
TD$(4,3)$ to give a BRGDD$_2(4,6^4)$.  Use a GDD$(4,3^5)$ as the
master with weight 2 and ingredient BRGDD$_2(4,2^4)$ to give a
BRGDD$_2(4,6^5)$.  Truncated transversal construction with
$n\not\in\{2,3,6,10\}$, and $0\leq m\leq n$ gives a GDD$(\{4,5\},n^4
m^1)$. Give the points of this design weight 6, and the above BRGDDs
to give a BRGDD$_2(4,(6n)^4 (6m)^1)$ which will give a
BRD$(6(4n+m)+1,4,2)$ by filling the groups with an additional point if
have BRD$(6n+1,4,2)$ and BRD$(6m+1,4,2)$s.  This construction will
give BRD$(6t+1,4,2)$ for all $t$, if we have BRD$(6n+1,4,2)$ for
$1\leq t\leq 15$ and $26\leq t\leq 27$. de~Launey has shown BRDs exist
when $6t+1$ is a prime power (CRC[IV.4.14]), so we just have to show
BRDs for $t=9,14,15$. When $6t+1=55$ (de~Launey PhD CRC[p245 [8]])
gives a direct construction. For $v=85=7*(13-1)+1$ and $v=91=7*13$ we
can form PBDs from TDs, and then construct BRDs by breaking the
blocks.


   Malcolm Greig
   Jan. 1999