Constructions Of Nearly Optimal Codebooks Based On Cyclotomic Classes Of Order Four

An codebook is a set of unit-norm complex vectors in.Codebooks with low value of maximum cross-correlation amplitude I_max?C? are widely used in the fields of CDMA system,quantum information processing and coding theory.A codebook is called optimal if I_max?C?meets the Welch bound.However,due to the stricter constraints and many limitations,thus it is hard to construct the optimal codebooks meeting the Welch bound with equality.A codebook is called asymptotically optimal if I_max?C?nearly meets the Welch bound.In this way,the constraints of codebooks are relaxed and the parameters selection are more flexible.The properties of asymptotically optimal codebooks and optimal codebooks are similar when the codewords in the codebooks are long enough.It is desirable to construct codebooks nearly meeting the Welch bound with equality.We use difference sets and almost difference sets to construct asymptotically optimal codebooks.Ding constructed asymptotically optimal codebooks by using almost difference sets based on fourth-order cyclotomic classes over finite fields.Hu and Wu used difference sets and the product of Abelian groups to construct asymptotically optimal codebooks.According to the above two methods,a new asymptotically optimal codebook is constructed by using the almost difference sets based on the fourth-order cyclotomic classes and the product of finite fields.This paper is divided into the following sections:The first part briefly introduces the research background of codebooks,the research status of related theories,and we sort out the existing research results.The second part introduces the basic knowledge of codebooks.First,we introduce the finite field.The asymptotically optimal codebooks mentioned in this paper are constructed based on finite fields.Secondly,we introduce the characters of the finite fields and some related properties.Then,we introduce the cyclotomic classes and cyclotomic number.Finally,we introduce the difference sets and almost difference sets,and sort out the difference sets and almost difference sets constructed by cyclotomic classes.The third part is our main results.First,we construct a new asymptotically optimal codebook by using the new almost difference set E=?D₁×?????????×D₂? defined by the fourth-order cyclotomic classes and the product of two finite fields F₁ and F₂.We verify that the maximum cross-correlation amplitude I_max?C? is asymptotically to tne Welch bound of the codebook.At last,we use the almost difference sets based on fourth-order cyclotomic classes and the product of arbitrary number of finite fields to construct a codebook,and verify the I_max?C? nearly meets the Welch bound when is large enough.