  Let BS(m,n) denote the set of base sequences (A;B;C;D), with A and B of
length m and C and D of length n. The base sequence conjecture (BSC) asserts
that BS(n+1,n) exist (i.e., are non-empty) for all n. This is known to be true
for n <= 36 and when n is a Golay number. We show that it is also true for n=37
and n=38. It is worth pointing out that BSC is stronger than the famous
Hadamard matrix conjecture. In order to demonstrate the abundance of base
sequences, we have previously attached to BS(n+1,n) a graph Gamma_n and
computed the Gamma_n for n <= 27. We now extend these computations and
determine the Gamma_n for n=28,...,35. We also propose a conjecture describing
these graphs in general.
