Best known lower bounds and their corresponding solutions for n=11 to 15 in A089676.
By Dmitry Kamenetsky, 5/06/2018

a(11) >= 24:
(1 1 0 0 0 0 0 0 0 0 0) (0 0 1 1 0 1 0 0 1 1 1) (0 0 0 0 1 1 0 0 0 0 0) (0 1 1 1 1 1 0 0 0 1 0) (1 1 0 1 0 1 0 1 0 1 0) (1 0 1 0 0 0 0 1 0 0 1) (1 0 0 1 1 0 0 0 1 1 1) (1 1 0 1 0 1 1 0 1 1 1) (1 0 0 0 1 1 1 1 1 1 1) (1 1 1 0 1 0 1 1 0 1 1) (0 1 1 1 1 0 1 1 1 0 1) (0 1 0 0 0 1 0 1 1 0 0) (1 0 1 0 1 1 0 0 1 0 1) (1 0 0 1 0 0 1 0 1 0 0) (1 0 1 0 0 1 1 0 1 1 0) (1 0 1 1 0 0 0 1 1 1 0) (0 0 0 0 1 0 1 0 1 1 0) (0 0 1 1 1 0 1 1 0 1 0) (1 0 0 1 1 1 1 1 0 0 0) (0 0 1 1 0 0 1 0 0 0 1) (1 1 1 1 1 1 1 0 0 0 1) (0 0 0 0 0 1 0 1 0 1 1) (1 1 1 0 1 0 0 0 1 1 0) (0 1 0 0 0 0 1 0 0 1 1) 

a(12) >= 32:
(0 0 1 0 1 1 0 1 1 0 1 0) (1 1 1 0 1 1 0 0 0 0 0 1) (1 1 0 0 1 0 1 0 1 1 1 1) (0 0 0 1 1 1 1 0 1 1 1 0) (0 1 0 1 1 1 0 1 1 1 0 1) (0 0 1 1 0 0 0 1 1 1 0 0) (0 0 0 0 0 1 0 1 0 0 0 1) (1 1 0 1 0 0 0 1 0 0 0 0) (1 1 0 1 0 1 1 0 1 0 0 1) (1 0 0 1 1 0 0 0 0 1 0 1) (0 1 0 0 1 0 0 0 1 0 0 0) (0 0 1 0 1 0 1 0 0 0 1 1) (1 1 1 1 0 1 1 1 1 1 1 0) (0 0 0 0 1 0 1 1 0 1 0 0) (0 1 1 0 0 1 0 0 1 1 1 1) (0 1 0 1 0 0 1 1 0 1 1 1) (1 0 1 1 0 1 0 0 0 0 1 0) (0 1 1 1 1 0 0 0 0 1 1 0) (0 1 1 0 0 0 1 1 1 0 0 1) (1 1 1 0 0 0 1 0 0 1 0 0) (0 1 1 1 1 1 1 1 0 0 0 0) (1 0 0 0 0 1 0 0 1 1 0 0) (0 0 0 1 0 0 0 0 1 0 1 1) (1 0 1 0 0 0 0 1 0 1 1 1) (1 0 0 1 1 1 1 1 0 0 1 1) (1 0 0 0 0 0 1 1 1 0 1 0) (1 0 1 1 1 0 1 0 1 0 0 0) (1 1 1 1 1 0 0 1 1 0 1 1) (1 1 0 0 1 1 0 1 0 1 1 0) (0 1 0 0 0 1 1 0 0 0 1 0) (0 0 1 1 0 1 1 0 0 1 0 1) (1 0 1 0 1 1 1 1 1 1 0 1)

a(13) >= 33:
(0 0 0 0 1 0 1 1 1 1 0 1 1) (0 1 1 1 0 0 1 0 0 0 1 0 0) (0 0 1 1 1 0 0 0 0 1 1 1 1) (1 1 0 1 0 1 0 1 1 1 1 1 0) (1 0 0 0 1 1 0 1 0 0 0 0 1) (0 0 0 1 0 0 0 0 1 1 0 1 0) (1 0 0 1 1 0 0 1 1 1 1 0 1) (1 1 0 0 1 0 1 0 1 0 0 0 0) (1 0 0 1 0 0 1 0 0 1 0 0 1) (1 1 1 0 0 0 0 0 0 0 0 1 0) (1 0 1 0 1 0 1 1 0 1 1 0 0) (0 1 1 0 0 1 1 0 1 1 0 0 0) (0 1 0 0 1 1 1 0 0 1 1 0 1) (0 0 0 1 0 1 0 1 0 1 1 0 0) (0 1 1 0 0 0 0 1 0 1 0 0 1) (0 0 0 1 0 1 1 1 1 0 0 0 1) (1 0 0 1 0 0 1 1 1 0 1 1 0) (1 1 1 1 1 1 1 0 1 0 1 0 1) (1 0 0 1 0 1 0 0 0 0 1 1 1) (1 0 1 0 0 0 1 0 1 1 1 1 1) (0 1 1 0 1 1 0 0 1 0 0 1 1) (0 1 0 1 1 1 1 0 0 0 0 1 0) (1 0 0 0 0 1 1 1 0 1 0 1 0) (0 0 0 0 1 0 0 1 0 0 1 1 0) (0 0 1 1 1 1 1 1 1 1 1 1 0) (0 1 0 0 0 0 0 0 1 0 1 0 1) (1 0 0 0 1 1 0 0 1 1 1 1 0) (1 1 1 1 1 0 1 1 0 0 0 1 1) (1 0 1 1 0 1 0 1 1 1 0 1 1) (0 0 1 0 0 1 1 1 0 0 1 1 1) (1 0 1 1 1 1 0 0 0 1 0 0 0) (1 0 1 0 0 1 0 1 1 0 1 0 0) (0 1 1 1 1 0 0 1 1 0 0 0 0)

a(14) >= 64:
(0 0 1 1 0 1 1 0 1 1 0 0 1 1) (1 1 0 0 0 1 1 0 0 1 0 0 1 1) (0 1 1 1 0 0 0 1 0 1 1 1 0 0) (1 0 1 0 0 1 1 0 0 1 1 1 0 1) (0 1 0 1 0 1 0 1 1 1 0 1 1 1) (1 0 1 1 0 0 0 1 1 1 0 1 0 1) (1 1 0 0 1 0 1 0 1 0 0 0 0 0) (0 1 0 0 0 0 0 1 0 0 1 0 1 1) (1 0 0 1 0 1 1 0 0 0 1 0 1 0) (0 0 0 1 0 0 0 1 0 1 0 0 1 0) (1 0 1 1 1 1 1 1 1 0 1 1 1 1) (1 1 1 1 0 1 1 1 1 1 1 0 0 1) (0 0 1 0 0 0 0 0 1 1 1 0 0 0) (0 1 0 0 1 0 1 1 0 1 0 1 1 0) (1 1 1 0 0 1 0 0 0 0 1 0 0 0) (1 1 1 1 1 1 0 1 1 0 0 1 0 0) (0 0 1 0 1 1 0 0 0 0 1 0 1 1) (0 0 1 0 1 1 0 1 1 1 0 1 1 0) (1 0 0 0 1 1 0 0 1 0 1 1 0 0) (0 1 0 0 1 1 0 1 1 1 1 0 0 0) (1 0 1 0 1 0 1 1 0 1 0 0 1 1) (1 0 0 0 0 0 0 0 0 0 0 0 0 1) (0 1 0 1 0 1 1 1 0 0 0 0 0 0) (0 1 1 0 0 1 1 0 1 1 0 1 0 0) (1 0 0 1 1 0 1 0 1 1 1 0 0 1) (1 1 0 0 0 0 0 0 1 1 1 1 0 1) (0 1 0 1 0 0 1 0 1 1 1 0 1 0) (0 0 0 1 0 0 0 0 1 0 1 1 1 1) (0 1 1 0 1 0 0 0 1 0 1 1 1 0) (1 0 0 0 0 0 1 0 1 1 0 1 1 0) (0 1 1 1 0 0 0 0 1 0 0 0 0 1) (0 0 0 1 1 1 0 1 1 0 0 0 0 1) (1 1 1 1 1 0 1 0 1 1 0 1 1 1) (0 0 1 0 1 0 1 0 1 0 0 1 0 1) (1 0 1 0 0 1 1 1 1 0 0 0 0 0) (1 0 0 1 1 0 1 1 0 0 0 1 0 0) (0 0 1 1 0 0 1 1 0 0 1 0 0 1) (1 1 0 0 0 1 1 1 1 0 1 1 1 0) (1 1 0 1 1 1 0 0 1 0 1 0 1 1) (0 0 1 1 0 1 0 0 0 0 0 1 0 0) (0 0 0 1 0 1 1 1 1 1 1 1 0 0) (1 0 1 1 1 0 0 0 1 0 0 0 1 0) (0 1 1 1 1 1 0 0 0 1 0 0 1 0) (0 0 0 0 1 0 0 1 0 1 1 1 0 1) (0 0 0 0 0 1 1 1 0 0 0 1 1 1) (1 0 1 1 1 1 0 1 0 1 1 0 0 0) (0 1 0 1 1 0 1 1 1 0 1 1 0 1) (1 1 0 1 0 0 0 0 0 0 0 1 1 0) (1 1 1 0 0 0 0 1 1 1 0 0 1 0) (1 0 0 1 1 1 0 0 0 1 0 1 1 1) (1 0 1 0 0 0 0 1 0 0 1 1 1 0) (0 1 1 0 1 1 1 1 1 0 0 0 1 1) (1 1 0 1 1 1 1 0 0 1 1 1 0 0) (0 1 0 0 1 1 0 0 0 0 0 1 0 1) (1 1 0 1 1 0 0 1 0 1 0 0 0 1) (1 1 1 0 1 1 0 1 0 1 1 1 1 1) (0 0 0 0 1 1 1 0 0 1 0 0 0 0) (1 0 0 0 0 1 0 1 1 1 1 0 1 1) (1 1 1 1 1 0 1 1 0 0 1 0 1 0) (0 1 1 0 1 0 1 0 0 1 1 0 0 1) (1 1 1 0 0 0 1 1 0 0 0 1 0 1) (0 0 0 0 1 0 1 1 1 0 1 0 1 0) (0 0 1 1 1 0 1 0 0 1 1 1 1 0) (0 1 1 1 0 1 1 0 0 0 1 1 1 1) 

a(15) >= 128:
(0 0 0 0 0 0 0 0 0 0 0 0 0 0 0) (0 0 0 0 0 0 1 0 1 1 0 1 1 1 0) (0 0 0 0 0 1 0 1 1 0 1 0 1 1 0) (0 0 0 0 0 1 1 0 0 0 1 1 1 0 1) (0 0 0 0 1 0 0 1 1 1 1 1 0 0 0) (0 0 0 0 1 0 1 0 0 1 1 0 0 1 1) (0 0 0 0 1 1 0 1 0 0 0 1 0 1 1) (0 0 0 0 1 1 1 1 1 1 0 0 1 0 1) (0 0 0 1 0 0 0 1 1 0 0 1 1 0 1) (0 0 0 1 0 0 1 1 0 0 1 1 0 1 0) (0 0 0 1 0 1 0 1 0 1 1 0 0 0 1) (0 0 0 1 0 1 1 0 1 0 0 0 0 1 1) (0 0 0 1 1 0 0 1 0 1 0 0 1 1 0) (0 0 0 1 1 0 1 0 1 0 1 0 1 0 0) (0 0 0 1 1 1 0 0 1 1 1 1 1 1 1) (0 0 0 1 1 1 1 0 0 1 0 1 0 0 0) (0 0 1 0 0 0 0 1 0 1 1 1 1 1 1) (0 0 1 0 0 0 1 1 1 0 1 0 0 0 1) (0 0 1 0 0 1 0 0 1 1 0 1 0 0 1) (0 0 1 0 0 1 1 1 0 1 0 0 0 1 0) (0 0 1 0 1 0 0 0 1 0 0 0 1 1 1) (0 0 1 0 1 0 1 1 0 0 0 1 1 0 0) (0 0 1 0 1 1 0 0 0 1 1 0 1 0 0) (0 0 1 0 1 1 1 0 1 0 1 1 0 1 0) (0 0 1 1 0 0 0 0 1 1 1 0 0 1 0) (0 0 1 1 0 0 1 0 0 1 0 0 1 0 1) (0 0 1 1 0 1 0 0 0 0 0 1 1 1 0) (0 0 1 1 0 1 1 1 1 1 1 1 1 0 0) (0 0 1 1 1 0 0 0 0 0 1 1 0 0 1) (0 0 1 1 1 0 1 1 1 1 0 1 0 1 1) (0 0 1 1 1 1 0 1 1 0 0 0 0 0 0) (0 0 1 1 1 1 1 1 0 0 1 0 1 1 1) (0 1 0 0 0 0 0 1 1 1 0 0 0 1 1) (0 1 0 0 0 0 1 1 0 1 1 0 1 0 0) (0 1 0 0 0 1 0 0 0 1 1 1 0 1 0) (0 1 0 0 0 1 1 1 1 0 0 1 0 0 0) (0 1 0 0 1 0 0 0 0 1 0 1 1 0 1) (0 1 0 0 1 0 1 1 1 0 1 1 1 1 1) (0 1 0 0 1 1 0 0 1 0 1 0 0 0 1) (0 1 0 0 1 1 1 0 0 0 0 0 1 1 0) (0 1 0 1 0 0 0 0 0 0 1 0 1 1 1) (0 1 0 1 0 0 1 0 1 1 1 1 0 0 1) (0 1 0 1 0 1 0 0 1 1 0 0 1 0 0) (0 1 0 1 0 1 1 1 0 1 0 1 1 1 1) (0 1 0 1 1 0 0 0 1 0 0 1 0 1 0) (0 1 0 1 1 0 1 1 0 0 0 0 0 0 1) (0 1 0 1 1 1 0 1 0 0 1 1 1 0 0) (0 1 0 1 1 1 1 1 1 1 1 0 0 1 0) (0 1 1 0 0 0 0 0 1 0 1 1 1 0 0) (0 1 1 0 0 0 1 0 0 0 0 1 0 1 1) (0 1 1 0 0 1 0 1 0 0 0 0 1 0 1) (0 1 1 0 0 1 1 0 1 1 1 0 1 1 1) (0 1 1 0 1 0 0 1 0 0 1 0 0 1 0) (0 1 1 0 1 0 1 0 1 1 0 0 0 0 0) (0 1 1 0 1 1 0 1 1 1 0 1 1 1 0) (0 1 1 0 1 1 1 1 0 1 1 1 0 0 1) (0 1 1 1 0 0 0 1 0 1 0 1 0 0 0) (0 1 1 1 0 0 1 1 1 0 0 0 1 1 0) (0 1 1 1 0 1 0 1 1 0 1 1 0 1 1) (0 1 1 1 0 1 1 0 0 0 1 0 0 0 0) (0 1 1 1 1 0 0 1 1 1 1 0 1 0 1) (0 1 1 1 1 0 1 0 0 1 1 1 1 1 0) (0 1 1 1 1 1 0 0 0 1 0 0 0 1 1) (0 1 1 1 1 1 1 0 1 0 0 1 1 0 1) (1 0 0 0 0 0 0 0 1 0 1 1 0 1 1) (1 0 0 0 0 0 1 1 0 1 0 1 0 0 1) (1 0 0 0 0 1 0 0 0 1 0 0 1 1 1) (1 0 0 0 0 1 1 0 1 1 1 0 0 0 0) (1 0 0 0 1 0 0 1 0 0 1 0 1 0 1) (1 0 0 0 1 0 1 1 1 0 0 0 0 1 0) (1 0 0 0 1 1 0 0 1 0 0 1 1 0 0) (1 0 0 0 1 1 1 1 0 1 1 1 1 1 0) (1 0 0 1 0 0 0 0 0 1 1 1 1 0 0) (1 0 0 1 0 0 1 1 1 1 1 0 1 1 1) (1 0 0 1 0 1 0 1 1 1 0 1 0 1 0) (1 0 0 1 0 1 1 1 0 0 0 0 1 0 0) (1 0 0 1 1 0 0 0 1 1 0 0 0 0 1) (1 0 0 1 1 0 1 0 0 0 0 1 1 1 1) (1 0 0 1 1 1 0 0 0 0 1 0 0 1 0) (1 0 0 1 1 1 1 1 1 0 1 1 0 0 1) (1 0 1 0 0 0 0 1 1 1 0 0 1 0 0) (1 0 1 0 0 0 1 0 0 0 1 0 1 1 0) (1 0 1 0 0 1 0 1 0 0 1 1 0 0 0) (1 0 1 0 0 1 1 1 1 0 0 1 1 1 1) (1 0 1 0 1 0 0 0 0 1 0 1 0 1 0) (1 0 1 0 1 0 1 0 1 1 1 1 1 0 1) (1 0 1 0 1 1 0 1 1 1 1 0 0 1 1) (1 0 1 0 1 1 1 0 0 0 0 0 0 0 1) (1 0 1 1 0 0 0 1 0 0 0 0 0 1 1) (1 0 1 1 0 0 1 0 1 0 0 1 0 0 0) (1 0 1 1 0 1 0 0 1 0 1 0 1 0 1) (1 0 1 1 0 1 1 0 0 1 1 1 0 1 1) (1 0 1 1 1 0 0 1 1 0 1 1 1 1 0) (1 0 1 1 1 0 1 1 0 1 1 0 0 0 0) (1 0 1 1 1 1 0 1 0 1 0 1 1 0 1) (1 0 1 1 1 1 1 0 1 1 0 0 1 1 0) (1 1 0 0 0 0 0 1 0 0 0 1 1 1 0) (1 1 0 0 0 0 1 0 1 0 0 0 1 0 1) (1 1 0 0 0 1 0 1 1 1 1 1 1 0 1) (1 1 0 0 0 1 1 1 0 0 1 0 0 1 1) (1 1 0 0 1 0 0 0 1 1 1 0 1 1 0) (1 1 0 0 1 0 1 0 0 0 1 1 0 0 0) (1 1 0 0 1 1 0 1 0 1 0 0 0 0 0) (1 1 0 0 1 1 1 0 1 1 0 1 0 1 1) (1 1 0 1 0 0 0 1 1 0 1 0 0 0 0) (1 1 0 1 0 0 1 0 0 1 0 0 0 1 0) (1 1 0 1 0 1 0 0 0 0 0 1 0 0 1) (1 1 0 1 0 1 1 0 1 0 1 1 1 1 0) (1 1 0 1 1 0 0 1 0 1 1 1 0 1 1) (1 1 0 1 1 0 1 1 1 1 0 1 1 0 0) (1 1 0 1 1 1 0 1 1 0 0 0 1 1 1) (1 1 0 1 1 1 1 0 0 1 1 0 1 0 1) (1 1 1 0 0 0 0 0 0 1 1 0 0 0 1) (1 1 1 0 0 0 1 1 1 1 1 1 0 1 0) (1 1 1 0 0 1 0 0 1 0 0 0 0 1 0) (1 1 1 0 0 1 1 0 0 1 0 1 1 0 0) (1 1 1 0 1 0 0 1 1 0 0 1 0 0 1) (1 1 1 0 1 0 1 1 0 1 0 0 1 1 1) (1 1 1 0 1 1 0 0 0 0 1 1 1 1 1) (1 1 1 0 1 1 1 1 1 0 1 0 1 0 0) (1 1 1 1 0 0 0 0 1 1 0 1 1 1 1) (1 1 1 1 0 0 1 1 0 0 1 1 1 0 1) (1 1 1 1 0 1 0 1 0 1 1 0 1 1 0) (1 1 1 1 0 1 1 1 1 1 0 0 0 0 1) (1 1 1 1 1 0 0 0 0 0 0 0 1 0 0) (1 1 1 1 1 0 1 0 1 0 1 0 0 1 1) (1 1 1 1 1 1 0 0 1 1 1 1 0 0 0) (1 1 1 1 1 1 1 1 0 0 0 1 0 1 0)