h H =[I-PT] h-k Example:For(7 4)Hamming code 「1101000 0110100 100:1011 Co=4+42+4 G 1110010 →H=0101110 G=+41+42 1010001 001011 C2=4+42+4 Thus,block code C=cGF(q)"H=0 3.Syndrome decoding Error vector (or error pattern):Let e be the transmitted codeword,and r be the received word.Then the difference between r and e gives the pattem of errors:e=r-c(for binary codes,.e=r⊕c) e:=1 indicates that the i-th position of r has an error ■Obviously.r=c+e There are in total 2"possible error patterns.Among them,only 2 patterns are correctable by an(n.k)li r code e To test whether a received vector r contains errors,we compute the following (n-k) tuple S=(so,S,.,5nmk-）erH =(c+e).H7 =cH"+eH"=eH 问题：s→e=? Ifs≠0→e≠0 fs=0→无错，c=0;或错误不可检：e∈C The (n-k)-tuple,s is called the syndrome of r. Any method solving these n-k equations is a decoding method. ■最小译码距离就是找重量最轻的e such that eH'=rH'=s Syndrome decoding consists of these steps: 1)Calculate syndrome s=rH"of received n-tuple. 2)Find最可能的错误图样e with eH'=s->非线性运算 17 0 1 T n k n k − − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = =− ⎡ ⎤ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ h h H IP h # # „ Example: For (7, 4) Hamming code N N 0 023 37 1 0 1 2 2 123 P 1101000 100 1011 0110100 010 1110 1110010 001 0111 1010001 T cuuu cuuu c uuu × ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎧ = + + ⎢ ⎥ ⎪ ⎢ ⎥ = ⇒ = ⇒ = ++ ⎢ ⎥ ⎨ ⎢ ⎥ ⎢ ⎥ ⎪ ⎢ ⎥ ⎩ = + + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ P G H # # # Thus, block code { ( ) } n C =∈ = c cH 0 GF q . 3. Syndrome decoding „ Error vector (or error pattern): Let c be the transmitted codeword, and r be the received word. Then the difference between r and c gives the pattern of errors: e=r-c (for binary codes, er c = ⊕ ) ej =1 indicates that the j-th position of r has an error. „ Obviously, r=c+e . „ There are in total 2n possible error patterns. Among them, only 2n-k patterns are correctable by an (n, k) linear code. „ To test whether a received vector r contains errors, we compute the following (n-k)- tuple: ( ) 01 1 , , T n k ss s = ⋅ − − s rH "  ( ) T = ceH + ⋅ TT T =+= cH eH eH 问题：s e ⇒ = ? If s0 e0 ≠⇒≠ If s0 e0 =⇒ = 无错， ； 或错误不可检：e∈C . „ The (n-k)-tuple, s is called the syndrome of r. Any method solving these n-k equations is a decoding method. „ 最小译码距离就是找重量最轻的 e such that T T eH rH s = = „ Syndrome decoding consists of these steps: 1) Calculate syndrome T s rH = of received n-tuple. 2) Find 最可能的错误图样 e with T eH s = -> 非线性运算