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تسجيل مستخدم جديدA New Method to Construct Gloay Complementary Set by Paraunitary Matrices and Hadamard Matrices
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Golay complementary sequences have been put a high value on the applications in orthogonal frequency-division multiplexing (OFDM) systems since its good peak-to-mean envelope power ratio(PMEPR) properties. However, with the increase of the code length, the code rate of the standard Golay sequences suffer a dramatic decline. Even though a lot of efforts have been paid to solve the code rate problem for OFDM application, how to construct large classes of sequences with low PMEPR is still difficult and open now. In this paper, we propose a new method to construct $q$-ary Golay complementary set of size $N$ and length $N^n$ by $Ntimes N$ Hadamard Matrices where $n$ is arbitrary and $N$ is a power of 2. Every item of the constructed sequences can be presented as the product of the specific entries of the Hadamard Matrices. The previous works in cite{BudIT} can be regarded as a special case of the constructions in this paper and we also obtained new quaternary Golay sets never reported in the literature.
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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 matric
es 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.
In this paper, we obtain a number of new infinite families of Hadamard matrices. Our constructions are based on four new constructions of difference families with four or eight blocks. By applying the Wallis-Whiteman array or the Kharaghani array to
the difference families constructed, we obtain new Hadamard matrices of order $4(uv+1)$ for $u=2$ and $vin Phi_1cup Phi_2 cup Phi_3 cup Phi_4$; and for $uin {3,5}$ and $vin Phi_1cup Phi_2 cup Phi_3$. Here, $Phi_1={q^2:qequiv 1pmod{4}mbox{ is a prime power}}$, $Phi_2={n^4in mathbb{N}:nequiv 1pmod{2}} cup {9n^4in mathbb{N}:nequiv 1pmod{2}}$, $Phi_3={5}$ and $Phi_4={13,37}$. Moreover, our construction also yields new Hadamard matrices of order $8(uv+1)$ for any $uin Phi_1cup Phi_2$ and $vin Phi_1cup Phi_2 cup Phi_3$.
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogona
lity over finite fields is an effective way to obtain good $q$-ary quantum codes of large length. Specifically, it follows from CSS construction (resp. Hermitian construction) that a matrix-product code over $mathbb{F}_{q}$ (resp. $mathbb{F}_{q^{2}}$) which is Euclidean dual-containing (resp. Hermitian dual-containing) can produce a $q$-ary quantum code. In order to obtain such matrix-product codes, a common way is to construct quasi-orthogonal matrices (resp. quasi-unitary matrices) as the defining matrices of matrix-product codes over $mathbb{F}_{q}$ (resp. $mathbb{F}_{q^{2}}$). The usage of NSC quasi-orthogonal matrices or NSC quasi-unitary matrices in this process enables the minimum distance lower bound of the corresponding quantum codes to reach its optimum. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in cite{Cao2020Constructioncaowang,Cao2020New,Cao2020Constructionof}; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and $q$-ary quantum codes derived from these newly constructed matrix-product codes.
This paper considers the problem of recovering an unknown sparse ptimes p matrix X from an mtimes m matrix Y=AXB^T, where A and B are known m times p matrices with m << p. The main result shows that there exist constructions of the sketching matric
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