Algebraic Coding 1993: Algebraic Coding pp 126-140

# On complementary sequences

• Amnon Gavish
• Abraham Lempel
Sequences
Part of the Lecture Notes in Computer Science book series (LNCS, volume 781)

## Abstract

A set of complex-valued sequences {Sj} j=1 k , where Sj =(sj1, sj2, ..., sjN) is called complementary if the sum R(·) of their auto-correlation functions {Rs j (·)} kj=1 k satisfies
$$R\left( \tau \right) = \sum\limits_{j = 1}^k {\sum\limits_{i = 1}^{N - \tau } {s_{ji} s_{ji}^* + \tau = 0} } ,\forall \tau \ne 0.$$

In this paper, we introduce a new family of complementary pairs of sequences over the alphabet α3={+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 α2={+1,−1}, and to new nontrivial constructions over the ternary alphabet α3. 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 α3. The bound is tight, as it is met by some of the proposed constructions, for infinitely many lengths.

## Keywords

Complementary sequences autocorrelation Golay pairs

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