On ternary complementary pairs (Q2712520)

For two finite sequences \(A= \{a_0,a_1,\dots, a_{\ell-1}\}\) and \(B= \{b_0,b_1,\dots, b_{\ell-1}\}\) of elements in \(\{-1,0,1\}\) their non-periodic autocorrelation \(N_{A,B}(s)\) is defined by NEWLINE\[NEWLINEN_{A,B}(s)= \sum^{\ell-1-s}_{i= 0} a_ia_{i+s}+ \sum^{\ell-1-s}_{i=0} b_i b_{i+s},\quad s= 0,1,\dots, \ell-1.NEWLINE\]NEWLINE If \(N_{A,B}(s)= 0\) for all \(s\), then \(A\), \(B\) is called a ternary complementary pair (TCP) of length \(\ell\). The weight \(w_{A,B}\) of the pair \(A\), \(B\) is the total number of non-zero entries in both \(A\) and \(B\). The authors give some new constructions for infinite families of TCP's. They also settle cases of existence or nonexistence of TCP's of lengths \(\ell\leq 20\) and weights \(\leq 40\). Optimal TCP's of all lengths \(\ell\) at most 22 are also given.