Construction Of Complementary Sets Research Articles

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

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.

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

The construction of complementary sets (CSs) of sequences with different set sizes and sequence lengths become important due to the practical application in orthogonal frequency-division multiplexing (OFDM) systems. Most of the constructions of CSs based on generalized Boolean functions (GBFs) are of length 2α (α is the non-negative integer). Recently, some works have reported on the construction of CSs having lengths non-power of two, i.e., in the form of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m-1</sup> + 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sup> (m is the non-negative integer, 0 ≤ v <; m). In this letter, we propose the constructions of CSs of lengths N +1 and N +2 for set size 4n, and CSs of length 2N + 3 for set size 8n, by employing the insertion method on Golay complementary pairs of length N. This systematic construction can generate more CSs of the new sequence length and set size, which has not been reported before.

A Novel Construction of Complementary Sets With Flexible Lengths Based on Boolean Functions

Golay complementary sets (GCSs) have been proposed to reduce the peak-to-average power ratios (PAPRs) in orthogonal frequency-division multiplexing (OFDM). They have upper bounds depending on the set sizes. The constructions of GCSs based on generalized Boolean functions have been proposed in the literature. However, most of these constructed GCSs have limited lengths, and hence they are not feasible for practical OFDM communication systems. This letter proposes a new construction of GCSs with flexible lengths. The proposed construction is a direct construction based on generalized Boolean functions. In addition, the constructed GCSs have various constellation sizes, set sizes, and upper bounds on PAPR.