A (92,8,2) design.


A difference family over $Z_{91}$ for a $(8,2)$ GDD of type $7^{13}$:
is

(5,6,7,10,22,42,63,90).

multiply the given block by 1, 9 and 81 to give 3 base blocks; the
groups are given by the residues modulo 13. Since the base blocks each
span all the non-zero residues modulo 13, the 7 shifts that have the
same residue modulo 13 form a holey 2-resolution set, hence this GDD
is a 2-frame.

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A (141,8,2) design


A difference family over $Z_7\times(Z_{19}\cup\{\infty\})$ for a
$(8,2)$ GDD of type $7^{20}$ is:

B_1 =  (0,   1,  11, 121,   4,  44,  85) 
B_2 =  (0,  10, 110,  13,  68,  83, 115)
B_3 =  (0,   1,  16,  22,  53,  70,  99, 125)


we represent $Z_7\times Z_{19}$ as $Z_{133}$.  Multiply the last block
by 1, 11 and 121 to give 5 base blocks; the first two (deficient)
blocks span the mod 7 residues twice, so we may add the infinite point
$\{x\}\times \{\infty\}$ to $B_1+y$, $B_2+y$ for $y\equiv x\bmod7$.



Malcolm Greig