Generalized Constructions of Complementary Sets of Sequences of Lengths Non-Power-of-Two

Abstract

The construction of complementary sets (CSs) of sequences with different set size and sequence length become important due to its practical application for OFDM systems. Most of the constructions of CSs, based on generalized Boolean functions (GBFs), are of length $2^\alpha$ ($\alpha$ is a natural number). Recently some works have been reported on construction of CSs having lengths non-power of two, i.e., in the form of $2^{m-1}+2^v$ ($m$ is natural number, $0\leq v <m $), $N+1$ and $N+2$, where $N$ is a length for which $q$-ary complementary pairs exist. In this paper, we propose a construction of CSs of lengths $M+N$ for set size $4n$, using concatenation of CSs of lengths $M$ and $N$, and set size $4n$, where $M$ and $N$ are lengths for which $q$-ary complementary pairs exists. Also, we construct CSs of length $M+P$ for set size $8n$ by concatenating CSs of lengths $M$ and $P$, and set size $8n$, where $M$ and $P$ are lengths for which $q$-ary complementary pairs and complementary sets of size $4$ exists, respectively. The proposed constructions cover all the previous constructions as special cases in terms of lengths and lead to more CSs of new sequence lengths which have not been reported before.