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Register a new userNew constructions of $q$-Ary 2-D Z-Complementary Array Pairs
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Added by
Sihem Mesnager
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This paper is devoted to sequences and focuses on designing new two-dimensional (2-D) Z-complementary array pairs (ZCAPs) by exploring two promising approaches. A ZCAP is a pair of 2-D arrays, whose 2-D autocorrelation sum gives zero value at all time shifts in a zone around the $(0,0)$ time shift, except the $(0,0)$ time shift. The first approach investigated in this paper uses a one-dimensional (1-D) Z-complementary pair (ZCP), which is an extension of the 1-D Golay complementary pair (GCP) where the autocorrelations of constituent sequences are complementary within a zero correlation zone (ZCZ). The second approach involves directly generalized Boolean functions (which are important components with many applications, particularly in (symmetric) cryptography). Along with this paper, new construction of 2-D ZCAPs is proposed based on 1-D ZCP, and direct construction of 2-D ZCAPs is also offered directly by 2-D generalized Boolean functions. Compared to existing constructions based on generalized Boolean functions, our proposed construction covers all of them. ZCZ sequences are a class of spreading sequences having ideal auto-correlation and cross-correlation in a zone around the origin. In recent years, they have been extensively studied due to their crucial applications, particularly in quasi-synchronous code division multiple access systems. Our proposed 2-D ZCAPs based on 2-D generalized Boolean functions have larger 2-D $mathrm{ZCZ}_{mathrm{ratio}}=frac{6}{7}$. Compared to the construction based on ZCPs, our proposed 2-D ZCAPs also have the largest 2-D $mathrm{ZCZ}_{mathrm{ratio}}$.
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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 $2times2timescdotstimes2$.
Cross Z-complementary pairs (CZCPs) are a special kind of Z-complementary pairs (ZCPs) having zero autocorrelation sums around the in-phase position and end-shift position, also having zero cross-correlation sums around the end-shift position. It can be utilized as a key component in designing optimal training sequences for broadband spatial modulation (SM) systems over frequency selective channels. In this paper, we focus on designing new CZCPs with large cross Z-complementary ratio $(mathrm{CZC}_{mathrm{ratio}})$ by exploring two promising approaches. The first one of CZCPs via properly cascading sequences from a Golay complementary pair (GCP). The proposed construction leads to $(28L,13L)-mathrm{CZCPs}$, $(28L,13L+frac{L}{2})-mathrm{CZCPs}$ and $(30L,13L-1)-mathrm{CZCPs}$, where $L$ is the length of a binary GCP. Besides, we emphasize that, our proposed CZCPs have the largest $mathrm{CZC}_{mathrm{ratio}}=frac{27}{28}$, compared with known CZCPs but no-perfect CZCPs in the literature. Specially, we proposed optimal binary CZCPs with $(28,13)-mathrm{CZCP}$ and $(56,27)-mathrm{CZCP}$. The second one of CZCPs based on Boolean functions (BFs), and the construction of CZCPs have the largest $mathrm{CZC}_{mathrm{ratio}}=frac{13}{14}$, compared with known CZCPs but no-perfect CZCPs in the literature.
Lattices have been used in several problems in coding theory and cryptography. In this paper we approach $q$-ary lattices obtained via Constructions D, $D$ and $overline{D}$. It is shown connections between Constructions D and $D$. Bounds for the minimum $l_1$-distance of lattices $Lambda_{D}$, $Lambda_{D}$ and $Lambda_{overline{D}}$ and, under certain conditions, a generator matrix for $Lambda_{D}$ are presented. In addition, when the chain of codes used is closed under the zero-one addition, we derive explicit expressions for the minimum $l_1$-distances of the lattices $Lambda_{D}$ and $Lambda_{overline{D}}$ attached to the distances of the codes used in these constructions.
Zero correlation zone (ZCZ) sequences and Golay sequences are two kinds of sequences with different preferable correlation properties. It was shown by Gong textit{et al.} and Chen textit{et al.} that some Golay sequences also possess a large ZCZ and are good candidates for pilots in OFDM systems. Known Golay sequences with ZCZ reported in the literature have a limitation in the length which is the form of a power of 2. One objective of this paper is to propose a construction of Golay complementary pairs (GCPs) with new lengths whose periodic autocorrelation of each of the Golay sequences and periodic corss-correlation of the pair displays a zero correlation zone (ZCZ) around the in-phase position. Specifically, the proposed GCPs have length $4N$ (where, $N$ is the length of a GCP) and ZCZ width $N+1$. Another objective of this paper is to extend the construction to two-dimensional Golay complementary array pairs (GCAPs). Interestingly the periodic corss-correlation of the proposed GACPs also have large ZCZs around the in-phase position.
The $q$-ary block codes with two distances $d$ and $d+1$ are considered. Several constructions of such codes are given, as in the linear case all codes can be obtained by a simple modification of linear equidistant codes. Upper bounds for the maximum cardinality of such codes is derived. Tables of lower and upper bounds for small $q$ and $n$ are presented.
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