由於完美的自我相關特性，格雷互補集合可以採用在正交頻域多工(orthogonal frequency division multiplexing) 系統上減少尖峰平均功率比(peak-to-average power ratio)。配合上完美的互相關特性，完全互補碼可以採用在分碼多工系統(code-division multiple access) 上削減其他使用者的訊號干擾。文獻中已知互補集合以及完全互補碼可以透過廣義里德穆勒碼建構而得。在本篇論文中，一個整合性的建構方法被提出，除了產生互補集合以及完全互補碼之外，同時可產生其他擁有特定的自相關及(或)互相關特性的互補集合家族。包含所謂的倍數平移互補集合，類互補集合，以及類完全互補碼。
本篇論文除了探討互補集合家族與里德穆勒碼之間的關係，更提出了直接由一階里德穆勒碼的共集來產生這些互補集合家族。對於這些互補集合家族成員，更提出了尖峰平均功率比的理論上限。
接著，完全互補碼與類完全互補碼的建構方法被提出。同樣可由一階里德穆勒碼來產生這類家族。此篇論文中，更將所建構出的類完全互補碼應用在蜂巢式正交分頻多工系統上。由於類完全互補碼擁有好的自相關與互相關特性以及低尖峰平均功率比，根據其所設計出的引導序列結構非常適合應用在基地台搜尋中。模擬結果發現此篇論文所設計的引導序列比蜂巢式正交分頻多工系統之一的全球互通微波存取(WiMAX) 系統所採用的引導序列擁有較佳的基地台搜尋效能。

Due to the ideal autocorrelation property, Golay complementary sets (GCSs) can be applied to orthogonal frequency division multiplexing (OFDM) systems for peak-to-average power ratio (PAPR) reduction. With the additional ideal cross-correlation property, complete complementary codes (CCCs), which consist of mutually orthogonal GCSs, can be employed in code-division multiple access (CDMA) systems to eliminate the multiple-access interference. It has been
shown that both GCSs and CCCs can be obtained from cosets of the first-order generalized Reed-Muller codes. In the thesis, a unified work to construct families of complementary sets and CCCs from cosets of the first-order generalized Reed-Muller codes is proposed. Besides generalizing some previous results on GCSs and CCCs, extensions of GCSs and CCCs which have some desirable (even though nonideal) autocorrelation and/or cross-correlation properties are investigated, namely, multiple-shift complementary sets (MSCSs), quasi-complementary sets (QCSs), and quasi complete complementary codes (QCCCs).

The relationship between the families of GCSs and generalized Reed-Muller codes is first investigated in the thesis. Direct generic constructions of GCSs, MSCSs, and
QCSs from cosets of the first-order generalized Reed-Muller codes are proposed. Upper bounds on PAPRs of families of GCSs are also exploited.

Then constructions of CCCs and QCCCs from generalized Reed-Muller codes are provided. A novel application of the constructed QCCCs is proposed in the thesis to employ them as the preamble sequences for cell search in cell-based OFDM systems, due to their good auto-correlation and cross-correlation properties as well as low PAPR values. Simulation results show that the proposed QCCC-based preambles outperform the preambles employed in the WiMAX system, both in terms of PAPR and cell search performance.

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