[Paper Review] Asymptotically Optimal Golay-ZCZ Sequence Sets with Flexible Length

知子 谷, Zhengchun Zhou+3 more|arXiv (Cornell University)|Dec 16, 2021

Wireless Communication Networks Research4 citations

TL;DR

This paper proposes two novel constructions of Golay-ZCZ sequence sets with flexible lengths, extending beyond the traditional power-of-two constraints. The first construction yields optimal binary Golay-ZCZ sequences of length $4N$ with ZCZ width $N$, while the second, based on $(M,M,N)$-complete complementary codes (CCCs), produces asymptotically optimal polyphase sequences of length $M^2N$ with ZCZ width $(M-1)N$, achieving near-optimal performance as $M$ increases.

ABSTRACT

Zero correlation zone (ZCZ) sequences and Golay complementary sequences are two kinds of sequences with different preferable correlation properties. Golay-ZCZ sequences are special kinds of complementary sequences which also possess a large ZCZ and are good candidates for pilots in OFDM systems. Known Golay-ZCZ sequences reported in the literature have a limitation in the length which is the form of a power of 2. In this paper, we propose two constructions of Golay-ZCZ sequence sets with new parameters which generalize the constructions of Gong et al. (IEEE Transaction on Communications 61(9), 2013) and Chen et al (IEEE Transaction on Communications 61(9), 2018). Notably, one of the constructions results in optimal binary Golay-ZCZ sequences, while the other results in asymptotically optimal polyphase Golay-ZCZ sequences as the number of sequences increases.

Motivation & Objective

• To overcome the limitation of existing Golay-ZCZ sequences being restricted to lengths that are powers of two.
• To develop new constructions that yield Golay-ZCZ sequence sets with larger ZCZ widths and more flexible sequence lengths.
• To enhance the availability of complete complementary codes (CCCs) with non-power-of-two parameters to increase design flexibility.
• To achieve optimality or asymptotic optimality in ZCZ width relative to the Tang-Fan-Matsufuji bound for polyphase sequences.

Proposed method

• Constructs Golay-ZCZ sequence sets of length $4N$ using a Golay complementary pair (GCP) of length $N$, achieving a ZCZ width of $N$.
• Employs $(M,M,N)$-complete complementary codes (CCCs) to generate Golay-ZCZ sequences of length $M^2N$ with ZCZ width $(M-1)N$.
• Introduces a new iterative construction of CCCs via the Kronecker product to expand the range of available CCC parameters.
• Uses inverse discrete Fourier transform (IDFT) matrices to structure the sequence sets and ensure desired correlation properties.
• Applies generalized Boolean functions (GBFs) as a foundational tool for sequence generation, extending prior constructions.
• Validates constructions through theoretical analysis and computer search for binary CCCs with parameters $(4,4,N)$ for $N = 3,5,7,11,13$.

Experimental results

Research questions

• RQ1 Can Golay-ZCZ sequence sets be constructed with lengths that are not powers of two, while maintaining large zero correlation zones?
• RQ2 Is it possible to design Golay-ZCZ sequences with ZCZ widths that increase with the number of sequences, rather than decrease?
• RQ3 How can complete complementary codes (CCCs) be extended to non-power-of-two parameters to enhance sequence design flexibility?
• RQ4 To what extent can the proposed constructions achieve optimality or asymptotic optimality relative to the Tang-Fan-Matsufuji bound?
• RQ5 Can iterative constructions based on Kronecker products generate new CCCs with previously unreported parameters?

Key findings

• The proposed construction for length $4N$ yields binary Golay-ZCZ sequences with ZCZ width $N$, achieving optimality as the optimality factor $C = 1$.
• For polyphase sequences of length $M^2N$, the ZCZ width is $(M-1)N$, and the optimality factor $C = (M-1)/M$, approaching 1 as $M \to \infty$, confirming asymptotic optimality.
• The construction of $(M,M,N)$-CCCs via Kronecker product enables the generation of new CCCs with non-power-of-two parameters, increasing design flexibility.
• Computer search identified new binary CCCs with parameters $(4,4,3)$, $(4,4,5)$, $(4,4,7)$, $(4,4,11)$, and $(4,4,13)$, which serve as valid seed codes.
• The resulting Golay-ZCZ sequences are suitable for uplink grant-free NOMA systems due to low PAPR and low coherence in spreading matrices.
• The proposed constructions generalize prior works by Gong et al. and Chen et al., extending their results beyond power-of-two lengths and improving ZCZ width scalability.