# On complementary sequences

## Abstract

*S*

_{j}}

_{j=1}

^{k}, where S

_{j}=(

*s*

_{j1}, s

_{j2}, ..., s

_{j}

*N*) is called complementary if the sum R(·) of their auto-correlation functions {

*Rs*

_{ j }(·)}

_{kj=1}

^{ k }satisfies

In this paper, we introduce a new family of complementary pairs of sequences over the alphabet α_{3}={+1,−1,0}. The inclusion of zero in the alphabet, which may correspond to a pause in transmission, leads both to a better understanding of the conventional binary case, where the alphabet is *α*_{2}={+1,−1}, and to new nontrivial constructions over the ternary alphabet *α*_{3}. For every length *N*, we derive restrictions on the location of the zero elements and on the form of the member sequences of the pair. We also derive a bound on the minimum number of zeros, necessary for the existence of a complementary pair of length *N* over *α*_{3}. The bound is tight, as it is met by some of the proposed constructions, for infinitely many lengths.

## Keywords

Complementary sequences autocorrelation Golay pairs## Preview

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