Base sequences of lengths n + 1 n + 1 , n + 1 n + 1 , n, n are constructed for all decompositions of 4 n + 2 4n + 2 into four squares for n = 19 , … , 24 n = 19, \ldots ,24 . The construction is achieved through an algorithm which is also presented. It is proved through an exhaustive search that Turyn sequences do not exist for n = 18 , … , 27 n = 18, \ldots ,27 ; since Turyn sequences cannot exist for n = 28 n = 28 or 29, the first unsettled case is n = 30 n = 30 .