Linear Codes Based On Matrix Space Over Finite Fields

Let F_q be a finite field with q elements.(1)Let D_r are the set of all m-order square matrices of rank r on F_q.and S is the set of all m-order skew-symmetric,which define a set of mappings W={?A|?A(X)=Tr(AX),X?D_r,A?S},Where Tr represents the trace of the matrix.A linear code is constructed according to the mapping mode of the elements in the set W C(r,m,q)={cA=(Tr(AX₁),Tr(AX₂),…,Tr(AX_t))| A?S,X_i?D_r,i=1,…,t},t is equal to the number of elements in our set D_r.In this paper,the second chapter mainly discusses the linear code C(1,m,q),C(2,m,q),C(1,m,q),C(2,m,q)parameters.(2)based on(1),Let F be the set of m × m global symmetric matrices on F_q,and H be m × m global symmetric matrices on IF_q.Based on the construction method of C(r,m,q)in Chapter 2,the following four linear codes are constructed:C1(m,q)={cA=(Tr(AX₁),Tr(AX₂),…,Tr(AX_t4))| A?S,Xi?S,i=1,…,t₄},C2(m,q)={cA=(Tr(AX₁),Tr(AX₂),…,Tr(AX_t5))| A?F,Xi?F,i=1,…,t₅},C3(m,q)={cA=(Tr(AX₁),Tr(AX₂),…Tr(AX_t6))| A?H,Xi?H,i=1,…,t₆},C4(m,q)={cA=(Tr(AX₁),Tr(AX₂),…,Tr(AX_t6))| A?S,Xi?H,i=1,…,t₆}.In this paper,the third chapter mainly discusses the linear code C₁(m,q),C₂(m,q),C₃(m,q),C₄(m,q)parameters.