Research On The Construction Of Several Classes Of Asymptotically Optimal Codebooks

A codebook C with parameter（N,K）is a set of N unit-form complex vectors in（?）^K,the maximum inner product cross-correlation valueI_max（C）of C is a criterion to evaluate its performance.Codebooks with a smallI_max（C）have applications in many fields.In 1974,Welch gave a lower bound ofI_max（C）,the so-called Welch bound.The codebooks which reach the Welch bound are called optimal codebooks.However,the construction of optimal codebooks is difficult.In recent years,the construction of asymptotically optimal codebook has attracted much attention because it is relatively easy to construct and is a good substitute for optimal codebooks in practice.The exponential sum over finite fields or finite rings is one of the most important tools in the construction of codebooks.The dissertation is devoted to constructing codebooks bases by use of exponential sums over Galois rings.The main results are stated as follows.Firstly,by using the basic properties of Galois rings,the relation between Jacobi sums and Gauss sums over Galois rings is given,and the modules of Jacobi sums are determined completely.In addition,the modules of partial Eisenstein sums are determined.Secondly,two classes of codebooks are constructed by using Gauss sums and extended Gauss sums over Galois rings,and it is proved that these codebooks are asymptotically optimal with respect to Welch bounds.Finally,by properly choosing multiplicative characters,two classes of codebooks are constructed by using Jacobi sums and extended Jacobi sums over Galois rings,and it is proved that these codebooks are asymptotically optimal with respect to Welch bounds.Compared with the existing results,the parameters of these codebooks constructed in this paper are new and flexible,and these codebooks can cover some related conclusions in finite fields.