The complete existence spectrum of periodic complementary ternary pairs of weight nine

Abstract

For two sequences with entries $0,\pm1$, periodic complementarity means that their periodic autocorrelation functions sum to zero at every nonzero shift. We give necessary and sufficient conditions on the length for the existence of such pairs of total weight nine, where weight counts nonzero entries. The conditions decide existence at any prescribed length by divisibility tests. The main new result bounds the length of primitive pairs with two nonzero components. A pair is primitive if it cannot be shortened by removing uniformly inserted zeros after independent cyclic permutations. We prove primitive-length bounds of $64,48,64$ for weight distributions $(5,4),(6,3),(7,2)$. These bounds make the classification finite and prove that the earlier searches contain every required case. Exhaustive enumeration gives $91$ primitive sequence-equivalence classes. Including the five known circulant classes gives $96$ representatives from which every pair can be generated. The same catalogue gives class counts at every length. For constructing one pair, $33$ displayed seeds suffice. The results settle earlier conjectures on the weight distributions and decide exactly which orders admit the associated two-circulant weighing-matrix construction.