| Similar to the study of typical groups,this paper constructs a group based on the unit circulant matrix and analyzes the structure of the group.At the same time,use this unit circulant matrix to construct a linear code and study its parameters.Let Fq be a finite field with q elements,Fqn to denote the n-dimensional row vector space over IFq and K be an identity circulant matrix of order n,that is,(?)where q=2m,n and m are positive integers,In-1is an identity matrix of order n-1.All matrices T satisfying TKT'=K on Fq form a group with respect to matrix multiplication,denoted by Gn.Let P be an s-dimensional vector subspace in Fqn,and use the same letter P as the matrix representation of the vector space P,that is,P is an s × n matrix with rank s,and its rows form basis of P.If the rank of PKP is 2r,then the vector space P is called(s,r)type subspace.Let Ms,rn be the set of all(s,r)type subspaces in Fqn.Using P?Ms,rn and Q?Ms,rn to mark rows and columns respectively,we construct the matrix Hs,r,n,m=(cij):(?)Take Hs,r,n,mas the check matrix to get the linear code over Fq,which is recorded as Cs,r,n,m.The main contents:the structure of the group Gn when n is odd and n=4;when s=1,r=0,the parameters of the linear codes Cs,r,n,m orresponding to n=3,4. |
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