On base and Turyn sequences

Abstract:

Base sequences of lengths $n + 1$, $n + 1$, n, n are constructed for all decompositions of $4n + 2$ into four squares for $n = 19, \ldots ,24$. 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 $n = 18, \ldots ,27$; since Turyn sequences cannot exist for $n = 28$ or 29, the first unsettled case is $n = 30$.