g0「0111001 G= =101010 g2」110001 The codeword for the message u=(101)is c=u-G=1-(011100)+0-(101010)+1-(110001)=(101101) An(n,k)linear systematic code is completely specified by an kxn generator matrix of the following form: go P0Po1.Po-k- 10.01 G- P0P.P- 01.0 =[P:I] 8-J Pk-l.0Pk-“Pk-m-k-i 00.1 Pmatris with P-or1 For example,the (63)code above is a systematic code: C=u G2=4+4 9=+ parity-check equations C0=4十42 2.Parity-check matrix An(n,k)linear code can also be specified by an(n-k)xn matrix H. Let c=(coc.c)be an n-tuple.Then e is a codeword iff c-H'=0=(00.0) The matrix H is called a parity-check matrix.By definition GH'=0 For an (n,k)linear systematic code with generator matrix G=[PI],the parity-check matrix is 1616 0 1 2 ⎡ ⎤ ⎡01 1 1 0 0⎤ ⎢ ⎥ ⎢ ⎥ = = 1 0 1 0 1 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢1 1 0 0 0 1⎥ ⎣ ⎦ ⎣ ⎦ g G g g The codeword for the message (1 ) u = 0 1 is c uG = ⋅ = ⋅ 1 1 1 0 0 + ⋅ 0 1 01 0 + ⋅ 1 0 0 0 1 = 0 10 01 11 1 ( ) ( ) ( ) ( 1 1 0 1) „ An (n, k) linear systematic code is completely specified by an k×n generator matrix of the following form: [ ] 0 00 01 0, 1 1 10 11 1, 1 1 1,0 1,1 1, 1 0 1 1 ij n k n k k k k k k nk k k identity matrix P matrix with p or g pp p g pp p G PI g pp p − − − − − − − − −− × = ⎡ ⎡ ⎤ 0 0⎤ ⎢ ⎥ ⎢ ⎥ 0 1 0 = = = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎢ 0 0 1⎦ " " " " # # # # " "     For example, the (6, 3) code above is a systematic code: 5 2 4 1 3 0 c u c u c u = = = 2 01 10 2 0 12 parity-check equations cuu cu u c uu = + ⎫ ⎪ = + ⎬ ⎪ = + ⎭ 2. Parity-check matrix „ An (n, k) linear code can also be specified by an (n-k)×n matrix H. Let ( ) 01 1 n cc c = − c " be an n-tuple. Then c is a codeword iff (00 0) T n k − cH 0 ⋅ = = "   个 The matrix H is called a parity-check matrix. By definition, T GH 0 = „ For an (n, k) linear systematic code with generator matrix G PI = [ k ] , the parity-check matrix is „