New Constructions Of Asymptotically Optimal Codebooks Via Cyclotomic Classes Of Order 8

A codebook C with parameter(N,K)is a set of n unit-norm complex vectors in CK.The maximum correlation amplitudes of a codebook can measure the quality of the codebook.Imax(C)represents the maximum absolute value of the inner product of two different codewords in a codebook.Codebooks with minimal possible Imax(C)are desirable in many applications.Welch presented the lower bound IWelch=(?)of Imax(C).When the maximum correlation amplitude Imax(C)reaches the Welch bound,we call C a maximum-Welch-bound-equality(MWBE)codebook.MWBE codebooks are widely used in the fields of code-division multiple-access systems,compressed sensing and so on.It is very difficult to construct the MWBE codebook.At present,there are only several types of MWBE codebooks reported in the literature.In recent years,asymptotically optimal codebooks have attracted widespread attention,since they can be constructed easily and Imax(C)can approach the Welch bound for large enough N.Thus,they are good alternatives in many applications.Hu and Wu constructed some asymptotically optimal codebooks by using the product of Abelian groups and difference sets.According to this method,a new asymptotically optimal codebook is constructed based on the product of finite fields and the cyclotomic classes of order 8.Chapter one first introduces the background knowledge of this subject,then introduces the related knowledge of codebook,and sums up the existing methods of constructing the MWBE codebooks and the methods of constructing asymptotically optimal codebook.Chapter two mainly introduces the basic knowledge related to this paper.Firstly,we give the knowledge of finite field,cyclotomic class and cyclotomic numbers,and concretely give the cyclotomic numbers of order 8 and its calculation formula.We calculate the cyclotomic numbers of order 8 in some specific finite fields.Then,we introduce the concepts of difference sets and almost difference sets and give some examples.At the end of this chapter,we introduce the almost difference sets used in this paper which are constructed by Ding Cunsheng using the cyclotomic classes of order 8 in finite fields.Chapter three is the main result of this paper.Firstly,we estimate the result of additive characteristics acting on the almost difference set D = C0(8,q)?C1(8,q)?C2(8,q)?C5(8,q).Then we construct the asymptotically optimal codebook via the product of finite fields and the cyclotomic classes of order 8.It is proved that the maximum correlation amplitudes of codebook constructed in this paper does approach to the Welch bound when the codeword is long enough.