| Over the past three decades,the world economy has boomed and the demand for wireless communication services has grown dramatically.Therefore,how to efficiently utilize frequency band resources has become an important direction for the development of wireless communication technologies.After that,a series of wireless communication technologies based on theories of time division,frequency division,and space division were developed and applied.Orthogonal Frequency Division Multiplexing(OFDM)is one such multi-carrier technique that has attracted extensive research attention from academia,researchers,and industry.OFDM multicarrier transmission is used in many important wireless standards such as 3GPP,LTE-A and 5G.However,the transmitted signal of OFDM exhibits a high PeakTo-Average Power Ratio(PAPR).This high PAPR reduces the efficiency of the high power amplifier(HPA)and degrades the performance of the system.The inherent error control capability and simplicity of implementation make the encoding method more promising for practical OFDM system design.Examples include Hamming,Golay,and Reed-Solomon in group codes,some of which can be used to reduce PAPR.The goal of reducing PMEPR in OFDM transmission can be achieved by using codes selected from sequences in complementary sequence sets,especially GCSs,which has good low peak-to-average ratio performance.However,the main disadvantage of this approach is that good PAPR comes at the cost of bitrate.Therefore,sequence design with a good mathematical structure(relative to search attempts),increasing the number of sequences and increasing the code rate are still the breakthrough points of coding technology.This thesis is inspired by the PU algorithm over QAM proposed by Budisin et.al.(2018IEEE Tran.Inf.Theory)and "three-stage construction process" proposed by Fiedlerd et.al.(2008 J.Combin,Theory(Series A)),and benefits from the recently proposed method of extracting GBF from the PU matrices proposed by Wang et.al.(2021 IEEE Tran.Inf.Theory).A system of constructing Golay sequences over QAM is established,which has the following achievements:1.Prove that all H-ary GAPs of size 2×2×· · ·×2 are "standard",that is,their algebraic normal form(ANF)satisfies a specific form.2.Developed the "three-stage construction process" of Fiedler et al.to make it suitable for QAM.The connection between the constructions over QAM based on PU matrices and that based on GBFs is established.The constructions based on PU matrices are diverse,and the constructions based on Boolean function are convenient for application and calculation of algebraic properties.This paper builds a bridge for the unification of the two construction schemes.3.Provide the GBFs of obtained Golay complementary sequences based on standard Golay sequence,which covers the construction of generalized cases I-V as a special case.Those sequences are presented by the combination of the quaternary standard GCSs and compatible offsets.The numbers of the proposed GCSs are equal to the product of the number of the quaternary standard GCSs and the number of the compatible offsets.By providing new compatible offsets based on the factorization of the integer q,we propose two new constructions of 4q-QAM GCSs,which have the generalized cases I-V as special cases.Li(2018 IEEE Tran.Inf.Theory)and Liu rt.al.(2018 IEEE Tran.Inf.Theory)showed that the numbers of offsets in the generalized cases Ⅰ-Ⅲ and Ⅳ-Ⅴ are a linear polynomial and a quadratic polynomial of m,respectively.Denote the number of prime factors of q counted with multiplicity by Ω(q).The number of new offsets in our first construction is lower bounded by a polynomial of m with degreeΩ(q).In particular,for q = 4,the number of new offsets in our first construction is 7times that in the general case Ⅳ-Ⅴ.The number of new offsets in our second construction is lower bounded by a polynomial of m with degree Ω(q)+ 1.For example,the new offsets in our two constructions for q = 6,whose number is bounded by a cubic polynomial.4.Provide the GBFs of obtained Golay complementary sequences based on non-standard Golay sequence.Those sequences are presented by the combination of the quaternary non-standard Golay sequences and non-linear offsets.This is a new construction compared to other previous literatures.For the open question raised by Li in 2008 IEEE Tran.Inf.Theory: whether it is possible to directly use non-standard Golay sequences and known Offsets to construct Golay sequences on QAM.Our research shows that the conclusion holds only in special cases,and the answer is no for general cases. |
