Research On Several Classes Of Error-correcting Codes And Codebooks With Good Performance

Error-correcting codes and codebooks are important topics in communication theory.This paper mainly presents some constructions of good error-correcting codes and optimal codebooks.These codes can be used in data storage,cellular telephone transmission,secret sharing schemes,strong regular graphs and so on.Optimal codebooks have applications in direct spread CDMA communications,unitary space-time modulations,multiple description coding over erasure channels and so on.The main contributions of this paper are the following:?1?Generalized Gray map.Assume that p is a prime and f?x?is a ???_p^k-valued function,we investigate a gereralized Gray map G on ???_p^k and use exponential sums to express the Hamming weight of G?f?x??,which generalizes a known result by Carlet.As an application,a family of nonlinear codes over_p is obtained from the generalized Gray map.We use the Weil-type exponential sums over Galois rings to provide a lower bound for the minimum distance of these codes.?2?Linear codes.Some linear codes from some special functions are constructed.In some cases,the Hamming weight distributions of these linear codes are determined.In particular,we find some optimal codes achieving the Griesmer bound with new parameters.Besides,we show that these linear codes can be used to construct secret scheme schemes with interesting access structures and strong regular graphs with new parameters.?3?Constant-weight codes.Based on trace and norm functions,a construction of q-ary codes over finite fields is given.From this construction,we not only obtain two classes of q-ary optimal constant-weight codes achieving the generalized Johnson bound I,but also derive a family of optimal constant-composition codes achieving the celebrated Luo-Fu-Vinck-Chen bound.As byproducts,some one-weight and two-weight linear codes are obtained.?4?Cyclic codes.We use Gauss sums to represent the Hamming weights of a class of cyclic codes whose duals have two zeroes.A lower bound of its minimum Hamming distance is determined.In some cases,we give the Hamming weight distributions of the cyclic codes.In particular,we obtain a class of three-weight optimal cyclic codes achieving the Griesmer bound,which generalizes a known result by Vega,and several classes of cyclic codes with only a few weights,which solve an open problem by Vega.?5?Complete weight distributions.Based on the theory of quadratic forms over finite fields,the complete weight distributions of two classes of cyclic codes are also investigated.We explicitly present the complete weight enumerators of the cyclic codes.Particularly,we solve an open problem proposed by Feng and Luo.Complete weight distributions can be applied to study authentication codes and the Walsh transform of monomial functions over finite fields.?6?Codebooks.Based on₄-valued quadratic forms satisfying some certain conditions,a new construction of complex codebooks is presented.Using this construction,we obtain some optimal codebooks achieving the Levenshtein bound.The codebooks have parameters(2^2m+2^m,2^m)and a small alphabet size 6.In particular,a set of generalized Boolean bent functions satisfying the conditions is constructed.As byproducts,some Boolean bent functions are derived.