The element a whose powers generate all the nonzero elements of GF(2")is called a primitive element of GF(2) VI.Binary Linear Block codes An (n,k)linear block code over GF(2)is simply a k-dim subspace of the vector the binary n-tuples In any linear code,the all-zero word,as the vector-space origion,is always a codeword ('.'if c is a codeword,then (-c)is also a codeword,so dose c+(-c)). The Hamming weight w(e)of a vector c is the number of nonzero components of c. Obviously,w(c)=du(c,0) The minimum Hamming weight of a code C is the smallest Hamming weight of any nonzero codeword of C. wmn=2i0。wn(c) For a linear code,d(cc)=d(0c-c)=d(0.c)=w(c) dan=min{da(0,c-cy水c,cy∈C,i≠j}=minw()c=w 1.Generator matrix A generator matrix for a linear block code C of length n and dimension k is any kxn matrix G whose rows form a basis for c. Every codeword is a linear combination of the rows of G. G 81081.8- g 8k-108k-l.8-ln 「go ■Encoding procedure:c=u.G=[4o,4,.ai]: 4g1 g- Example:For a(6,3)linear block code, 15 15 „ The element α whose powers generate all the nonzero elements of ) (2m GF is called a primitive element of ) (2m GF . VI. Binary Linear Block codes „ An (n, k) linear block code over GF(2) is simply a k-dim subspace of the vector space Vn of all the binary n-tuples. „ In any linear code, the all-zero word, as the vector-space origion, is always a codeword (∵if c is a codeword, then (-c) is also a codeword，so dose c+(-c)). „ The Hamming weight w(c) of a vector c is the number of nonzero components of c. Obviously, H w d () (, ) c c0 = . „ The minimum Hamming weight of a code C is the smallest Hamming weight of any nonzero codeword of C. min , 0 min ( ) w w H ∈ ≠ = c c c C For a linear code, 12 2 1 ( , ) (, ) (,) () HH H d d dw c c 0c c 0c c = −= = min { ( ) } min 0 min , , , min ( ) H i j ij d d ij w w ≠ = − ∈ ≠= = c 0c c c c c C 1. Generator matrix „ A generator matrix for a linear block code C of length n and dimension k is any k×n matrix G whose rows form a basis for C. Every codeword is a linear combination of the rows of G. 0 00 01 0, 1 1 10 11 1, 1 1 1,0 1,1 1, 1 n n k k k kn k n gg g gg g gg g − − − − − −− × ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ g g G g . . # # . „ Encoding procedure: [ ] 0 1 1 01 1 0 1 , , k k ll l k uu u u − − = − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ =⋅ = = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ∑ g g c uG g g " # Example: For a (6, 3) linear block code