The Constructions Of Orbit Codes And Codebooks Based On Geometry Of Classical Groups

In random linear network coding,constant dimensional codes are very effective for transmitting messages.The special features of constant dimensional codes are that codewords correspond to subspaces and all the codewords have the same fixed dimension.The study of constant dimensional codes is a central problem in random linear network coding.The orbit codes are subsets of constant dimension code.They are defined as orbits of a subgroup of the general linear group acting on the Grassmannian.Codebooks are usually used to distinguish among the signals of different users in code-division-multiple-access communication system.In practical applications,codebooks with small inner-product correlation are required.One important problem is to minimize the maximal cross-correlation amplitude of codebooks.All kinds of typical spaces in the geometry of classical group over finite fields have good combinatorical structures and count easily,therefore,the typical spaces could be used to construct orbit codes and codebooks.Firstly,in this paper,we construct the orbit code based on the subspaces of type(m,k)in singular linear space over finite fields.Meanwhile,we give the parameters and basic properties of the constructed orbit codes and characterize the dual code and isometric orbit codes of the constructed orbit codes.Secondly,construction of codebooks is given by using vector spaces over finite fields.The construction produces the complex codebooks asymptotically achieving the Welch bound and Levenshtein bound.Our methods yield many codebooks with new parameters compared to the previous codebooks.Finally,construction of complex codebooks is provided by projective space over finite fields.A family of codebooks are obtained based on the construction.The codebooks asymptotically meet the Welch bound under some conditions.The parameters of the codebooks presented in this paper are new compared to the previous codebooks.