A Direct Construction of Optimal Symmetrical Z-Complementary Code Sets of Prime Power Lengths

This paper presents a direct construction of an optimal symmetrical Z-complementary code set (SZCCS) of prime power lengths using a multi-variable function (MVF). SZCCS is a natural extension of the Z-complementary code set (ZCCS), which has only front-end zero correlation zone (ZCZ) width. SZCCS has both front-end and tail-end ZCZ width. SZCCSs are used in developing optimal training sequences for broadband generalized spatial modulation systems over frequency-selective channels because they have ZCZ width on both the front and tail ends. The construction of optimal SZCCS with large set sizes and prime power lengths is presented for the first time in this paper. Furthermore, it is worth noting that several existing works on ZCCS and SZCCS can be viewed as special cases of the proposed construction

Root Cross Z-Complementary Pairs with Large ZCZ Width

In this paper, we present a new family of cross $Z$-complementary pairs (CZCPs) based on generalized Boolean functions and two roots of unity. Our key idea is to consider an arbitrary partition of the set $\{1,2,\cdots, n\}$ with two subsets corresponding to two given roots of unity for which two truncated sequences of new alphabet size determined by the two roots of unity are obtained. We show that these two truncated sequences form a new $q$-ary CZCP with flexible sequence length and large zero-correlation zone width. Furthermore, we derive an enumeration formula by considering the Stirling number of the second kind for the partitions and show that the number of constructed CZCPs increases significantly compared to the existing works.Comment: This work has been presented in 2022 IEEE International Symposium on Information Theory (ISIT), Espoo, Finlan

New Spectrally Constrained Sequence Sets With Optimal Periodic Cross-Correlation

Spectrally constrained sequences (SCSs) play an important role in modern communication and radar systems operating over non-contiguous spectrum. Despite numerous research attempts over the past years, very few works are known on the constructions of optimal SCSs with low cross-correlations. In this paper, we address such a major problem by introducing a unifying framework to construct unimodular SCS families using circular Florentine rectangles (CFRs) and interleaving techniques. By leveraging the uniform power allocation in the frequency domain for all the admissible carriers (a necessary condition for beating the existing periodic correlation lower bound of SCSs), we present a tighter correlation lower bound and show that it is achievable by our proposed SCS families including multiple SCS sets with zero correlation zone properties

Asymptotically Locally Optimal Weight Vector Design for a Tighter Correlation Lower Bound of Quasi-Complementary Sequence Sets

A quasi-complementary sequence set (QCSS) refers to a set of two-dimensional matrices with low nontrivial aperiodic auto- and cross-correlation sums. For multicarrier code-division multiple-access applications, the availability of large QCSSs with low correlation sums is desirable. The generalized Levenshtein bound (GLB) is a lower bound on the maximum aperiodic correlation sum of QCSSs. The bounding expression of GLB is a fractional quadratic function of a weight vector w and is expressed in terms of three additional parameters associated with QCSS: the set size K, the number of channels M, and the sequence length N. It is known that a tighter GLB (compared to the Welch bound) is possible only if the condition M ≥ 2 and K ≥ K̅ + 1, where K̅ is a certain function of M and N, is satisfied. A challenging research problem is to determine if there exists a weight vector that gives rise to a tighter GLB for all (not just some) K ≥ K̅ + 1 and M ≥ 2, especially for large N, i.e., the condition is asymptotically both necessary and sufficient. To achieve this, we analytically optimize the GLB which is (in general) nonconvex as the numerator term is an indefinite quadratic function of the weight vector. Our key idea is to apply the frequency domain decomposition of the circulant matrix (in the numerator term) to convert the nonconvex problem into a convex one. Following this optimization approach, we derive a new weight vector meeting the aforementioned objective and prove that it is a local minimizer of the GLB under certain conditions