On the base sequence conjecture

Let B S ( 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 B S ( 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. er to demonstrate the abundance of base sequences, we have previously attached to B S ( n + 1 , n ) a graph Γ n and computed the Γ n for n ≤ 27 . We now extend these computations and determine the Γ n for 28 ≤ n ≤ 35 . We also propose a conjecture describing these graphs in general.