[Paper Review] New Construction of Z-Complementary Code Sets and Mutually Orthogonal Complementary Sequence Sets

Bingsheng Shen, Yang Yang+1 more|arXiv (Cornell University)|May 21, 2021

Wireless Communication Networks Research30 references4 citations

TL;DR

This paper presents a direct construction of optimal $q$-ary Z-complementary code sets (ZCCSs) with parameters $(q^{v+1}, q, q^m, q^{m-v})$ using extended Boolean functions (EBFs), breaking the prior limitation of power-of-two parameters. It further generalizes existing complete complementary code (CCC) and mutually orthogonal complementary sequence set (MOCSS) constructions, enabling new sequence lengths—particularly for binary MOCSSs with $M/N = 1/2$—including previously unachievable lengths like 11 and 27.

ABSTRACT

Due to the zero nontrivial aperiodic correlation of complete complementary code (CCCs), it is used in asynchronous multi carrier code division multiple access (MC-CDMA) communication to provide zero interference performance. However, there is no CCC for some lengths. In this case, if the system pays more attention to the correlation, it can use mutually orthogonal complementary sequence sets (MOCSSs), and if the system pays more attention to the set size, it can use Z-complementary code sets (ZCCSs). In this paper, we propose a direct construction of $q$-ary $(q^{v+1},q,q^m,q^{m-v})$-ZCCS. In addition, based on the known CCCs, we propose a new construction of MOCSSs and CCCs respectively.

Motivation & Objective

• To address the limitation of existing Z-complementary code sets (ZCCSs) that are restricted to power-of-two parameters, especially for non-binary and flexible-length systems.
• To develop a direct construction of optimal $q$-ary ZCCSs that overcomes the dependency on second-order generalized Boolean functions (GBFs) and extends to arbitrary $q$.
• To generalize prior CCC and MOCSS constructions to allow for more flexible sequence lengths, particularly for systems with $M/N = 1/2$ ratio.
• To enable the construction of new MOCSSs with previously unattainable lengths, such as 11 and 27, by leveraging CCC-based concatenation techniques.

Proposed method

• Utilizes extended Boolean functions (EBFs), a generalization of Boolean functions mapping $\mathbb{Z}_q^m \to \mathbb{Z}_q$, to construct $q$-ary ZCCSs directly.
• Employs a novel construction of $q$-ary $(q^{v+1}, q, q^m, q^{m-v})$-ZCCSs, where $v \leq m$, ensuring optimality via zero correlation zone (ZCZ) properties.
• Generalizes the result of [17] to construct complete complementary codes (CCCs) from existing CCCs using a product-like operation on sequences.
• Proposes a concatenation-based method to generate $(M, 2M, L_1 + L_2)$-MOCSSs from two $(M, L_1)$- and $(M, L_2)$-CCCs, preserving orthogonality and zero cross-correlation.
• Applies interleaving and phase-shifting operations ($\mathbf{a} \oplus d$) to maintain correlation properties across sequences in the constructed sets.
• Validates the construction through analytical proof of zero aperiodic cross- and auto-correlation within the ZCZ and across all non-zero time shifts.

Experimental results

Research questions

• RQ1 Can a direct construction of optimal $q$-ary ZCCSs be achieved without relying on second-order generalized Boolean functions?
• RQ2 Can the parameter constraints of existing ZCCS constructions be relaxed to include non-power-of-two lengths and arbitrary $q$-ary alphabets?
• RQ3 Can existing CCC and MOCSS constructions be generalized to allow for more flexible sequence lengths, especially for $M/N = 1/2$ systems?
• RQ4 What new sequence lengths can be achieved using the proposed MOCSS construction compared to prior works?

Key findings

• The proposed $q$-ary ZCCS construction achieves optimal parameters $(q^{v+1}, q, q^m, q^{m-v})$ for any $q$, breaking the prior restriction to power-of-two parameters.
• The construction is direct and based on EBFs, enabling new ZCCS parameters not achievable with previous GBF-based methods.
• The paper constructs a binary $(2,4,11)$-MOCSS, a length not achievable by [16] or [17], demonstrating new flexibility in sequence length.
• A new binary $(2,4,27)$-MOCSS is constructed, which is not possible with prior constructions, highlighting the broader length coverage.
• The generalized CCC construction allows for the creation of new CCCs from existing ones, extending the applicability of known CCC families.
• The MOCSS construction achieves $M/N = 1/2$ ratio with significantly expanded length support, including all lengths $L \leq 40$ except those excluded by prior works.