Further Results on Base Sequences, Disjoint Complementary Sequences, \(OD(4t;t,t,t,t)\) and the Excess of Hadamard Matrices

We obtain new base sequences, that is four sequences of lengths \(m + p\), \(m + p\), \(m\), \(m\), with \(p\) odd, which have zero auto correlation function which can be used with Yang numbers and four disjoint complementary sequences (and matrices) with zero non-periodic (periodic) auto correlation function to form longer sequences.

We give an alternate construction for \(T\)-sequences of length \((4n + 3)(2m + p)\), where \(n\) is the length of a Yang nice sequence.

These results are then used in the Goethals-Seidel or (Seberry) Wallis-Whiteman construction to determine eight possible decompositions into squares of \((4n + 3)(2m + p)\) in terms of the decomposition into squares of \(2m + 1\) when there are four suitable sequences of lengths \(m + 1\), \(m + 1\), \(m\), \(m\) and \(m\), the order of four Williamson type matrices. The new results thus obtained are tabulated giving \({OD}(4t; t, t, t, t)\) for the new orders \(t \in \{121, 135, 217, 221, 225, 231, 243, 245, 247,\\ 253, 255, 259, 261, 265, 273, 275, 279, 285, 287, 289, 295, 297, 299\}\).

The Hadamard matrix with greatest known excess for these new \(t\) is then listed.