Constructions Of Several Classes Of Projective Linear Codes And Almost Optimal Codebooks

Projective linear codes and codebooks are important research topics in communication theory.Projective codes have important applications in computer storage,secret sharing schemes,combinatorial mathematics,etc.Codebooks with low correlation are widely used in radar communication,compressed sensing,etc.In this paper,we mainly use algebraic and number theoretic tools to construct several families of projective codes with good parameters and almost optimal codebooks.The main results are as follows:Firstly,based on the defining-set method,two families of projective linear codes are constructed by quadratic multiplicative characters.Their parameters and weight distributions are explicitly calculated.It turns out that the first family of linear codes is projective and has three weights.The second family of linear codes is also projective and has two weights.The duals are also almost optimal according to the sphere-packing bound.Besides,some selforthogonal codes and minimal codes are obtained.The self-orthogonal codes can be used to construct quantum codes and minimal codes can be used to construct secret sharing schemes with interesting structures.Secondly,two constructions of linear codes are presented with defining sets given by the intersection and difference of two sets.These constructions produce several families of new projective two-weight or three-weight linear codes.The parameters and weight distributions of linear codes are given.It turns out that the duals are optimal according to the sphere-packing bound.In addition,they have good applications in secret sharing schemes with interesting access structures,strong regular graphs,association schemes and so on.Thirdly,based on the cyclic codes constructed by Vega et al.,we study the weight distribution and parameters of the augmented codes of the cyclic codes.It is proved that the augmented code is projective.In particular,the augmented codes have higher code rate than that of the original codes.The duals of the augmented codes are almost optimal according to the sphere-packing bound.In addition,the complete weight distribution of the original codes is also determined.Fourthly,the absolute values or explicit values of the Hybrid character sums are determined under certain conditions.The distribution of correlation values and the maximum correlation magnitude of several partial Hadamard codebooks are studied by these Hybrid character sums.It turns out that the maximum magnitude of each codebook asymptotically achieves the Welch bound equality.In addition,the constructed codebooks are equivalent to nearly equiangular tight frames and have nice application in compressed sensing matrices.