Constructions Of Two Kinds Of Linear Codes Based On Polynomial And Combinatorial Design

Linear codes,as a very important class of codes,have a good algebraic structure in the field of incoding,which are the basis of discussing all kinds of codes.Polynomial codes and LDPC codes are two kinds of common linear codes.In this paper,polynomial codes and LDPC codes are constructed by combining finite fields with polynomials,finite geometry with combinatorial design,respectively.In the construction part of polynomial codes,firstly,a class of cyclic permutations is given based on positive integer sets,on which an equivalent relation is defined.Meanwhile,a kind of special polynomial functions is constructed by using this equivalence relation,and the linear space V_<sub>m,_s sover F_q is generated based on such linearly independent polynomial functions.Secondly,a class of polynomial codes is constructed in linear space V_<sub>m,_s s,and the relevant parameters of the codes are calculated.Furthermore,in order to get more codes,new parameters l is introduced and a similar method is discussed.Finally,another class of polynomial codes is constructed and their related parameters are calculated.In the construction part of LDPC codes,firstly,based on the orthogonal space over finite fieldF_q,a class of Steiner 3-designs is constructed and map it isomorphically onto a finite number set.Then,by redefining the incidence relation of 3-designs and their residual designs after isomorphism,higher incidence matrices are obtained.Therefore,two classes of binary regular LDPC codes are constructed through using the incidence matrices as the check matrices of LDPC codes.Finally,some concrete examples of these two classes of LDPC codes are given and simulated.