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={σA|σA（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.