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

Abstract

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.