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Register a new userConstructions of complementary sequence sets and complete complementary codes by 2-level autocorrelation sequences and permutation polynomials
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Zilong Wang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this paper, a recent method to construct complementary sequence sets and complete complementary codes by Hadamard matrices is deeply studied. By taking the algebraic structure of Hadamard matrices into consideration, our main result determine the so-called $delta$-linear terms and $delta$-quadratic terms. As a first consequence, a powerful theory linking Golay complementary sets of $p$-ary ($p$ prime) sequences and the generalized Reed-Muller codes by Kasami et al. is developed. These codes enjoy good error-correcting capability, tightly controlled PMEPR, and significantly extend the range of coding options for applications of OFDM using $p^n$ subcarriers. As another consequence, we make a previously unrecognized connection between the sequences in CSSs and CCCs and the sequence with 2-level autocorrelation, trace function and permutation polynomial (PP) over the finite fields.
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A new method to construct $q$-ary complementary sequence sets (CSSs) and complete complementary codes (CCCs) of size $N$ is proposed by using desired para-unitary (PU) matrices. The concept of seed PU matrices is introduced and a systematic approach on how to compute the explicit forms of the functions in constructed CSSs and CCCs from the seed PU matrices is given. A general form of these functions only depends on a basis of the functions from $Z_N$ to $Z_q$ and representatives in the equivalent class of Butson-type Hadamard (BH) matrices. Especially, the realization of Golay pairs from the our general form exactly coincides with the standard Golay pairs. The realization of ternary complementary sequences of size $3$ is first reported here. For the realization of the quaternary complementary sequences of size 4, almost all the sequences derived here are never reported before. Generalized seed PU matrices and the recursive constructions of the desired PU matrices are also studied, and a large number of new constructions of CSSs and CCCs are given accordingly. From the perspective of this paper, all the known results of CSSs and CCCs with explicit GBF form in the literature (except non-standard Golay pairs) are constructed from the Walsh matrices of order 2. This suggests that the proposed method with the BH matrices of higher orders will yield a large number of new CSSs and CCCs with the exponentially increasing number of the sequences of low peak-to-mean envelope power ratio.
A new method to construct $q$-ary complementary sequence (or array) sets (CSSs) and complete complementary codes (CCCs) of size $N$ is introduced in this paper. An algorithm on how to compute the explicit form of the functions in constructed CSS and CCC is also given. A general form of these functions only depends on a basis of functions from $Z_N$ to $Z_q$ and representatives in the equivalent class of Butson-type Hadamard matrices. Surprisingly, all the functions fill up a larger number of cosets of a linear code, compared with the existing constructions. From our general construction, its realization of $q$-ary Golay pairs exactly coincides with the standard Golay sequences. The realization of ternary complementary sequences of size $3$ is first reported here. For binary and quaternary complementary sequences of size 4, a general Boolean function form of these sequences is obtained. Most of these sequences are also new. Moreover, most of quaternary sequences cannot be generalized from binary sequences, which is different from known constructions. More importantly, both binary and quaternary sequences of size 4 constitute a large number of cosets of the linear code respectively.
Previously, we have presented a framework to use the para-unitary (PU) matrix-based approach for constructing new complementary sequence set (CSS), complete complementary code (CCC), complementary sequence array (CSA), and complete complementary array (CCA). In this paper, we introduce a new class of delay matrices for the PU construction. In this way, generalized Boolean functions (GBF) derived from PU matrix can be represented by an array of size $2times 2 times cdots times 2$. In addition, we introduce a new method to construct PU matrices using block matrices. With these two new ingredients, our new framework can construct an extremely large number of new CSA, CCA, CSS and CCC, and their respective GBFs can be also determined recursively. Furthermore, we can show that the known constructions of CSSs, proposed by Paterson and Schmidt respectively, the known CCCs based on Reed-muller codes are all special cases of this new framework. In addition, we are able to explain the bound of PMEPR of the sequences in the part of the open question, proposed in 2000 by Paterson.
The previous constructions of quadrature amplitude modulation (QAM) Golay complementary sequences (GCSs) were generalized as $4^q $-QAM GCSs of length $2^{m}$ by Li textsl{et al.} (the generalized cases I-III for $qge 2$) in 2010 and Liu textsl{et al.} (the generalized cases IV-V for $qge 3$) in 2013 respectively. Those sequences are presented as the combination of the quaternary standard GCSs and compatible offsets. By providing new compatible offsets based on the factorization of the integer $q$, we proposed two new constructions of $4^q $-QAM GCSs, which have the generalized cases I-V as special cases. The numbers of the proposed GCSs (including the generalized cases IV-V) are equal to the product of the number of the quaternary standard GCSs and the number of the compatible offsets. For $q=q_{1}times q_{2}times dotstimes q_{t}$ ($q_k>1$), the number of new offsets in our first construction is lower bounded by a polynomial of $m$ with degree $t$, while the numbers of offsets in the generalized cases I-III and IV-V are a linear polynomial of $m$ and a quadratic polynomial of $m$, respectively. In particular, the numbers of new offsets in our first construction is seven times more than that in the generalized cases IV-V for $q=4$. We also show that the numbers of new offsets in our two constructions is lower bounded by a cubic polynomial of $m$ for $q=6$. Moreover, our proof implies that all the mentioned GCSs over QAM in this paper can be regarded as projections of Golay complementary arrays of size $2times2timescdotstimes2$.
The concept of paraunitary (PU) matrices arose in the early 1990s in the study of multi-rate filter banks. So far, these matrices have found wide applications in cryptography, digital signal processing, and wireless communications. Existing PU matrices are subject to certain constraints on their existence and hence their availability is not guaranteed in practice. Motivated by this, for the first time, we introduce a novel concept, called $Z$-paraunitary (ZPU) matrix, whose orthogonality is defined over a matrix of polynomials with identical degree not necessarily taking the maximum value. We show that there exists an equivalence between a ZPU matrix and a $Z$-complementary code set when the latter is expressed as a matrix with polynomial entries. Furthermore, we investigate some important properties of ZPU matrices, which are useful for the extension of matrix sizes and sequence lengths. Finally, we propose a unifying construction framework for optimal ZPU matrices which includes existing PU matrices as a special case.
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