3)估计发送码字c=r-c 4)Determin uation c=iG is received.Then s=(so.552)=rHT=(111) Let e=(eo.e.)be the error pattern.Since s=eH',we have the following 3 equations 1=eote;+este6 1-er+es+e+es l=extertes+e6 There are 16 possible solutions,其中e=(O00O01O)是重量最小，是最可能发生的错误图 样，故c=r©e=(10010010)④(0000010)=(1001011) 。Standard array c1=0c2 |c3.cw e十C2e3十C3 e. e,++c3 M=2,r=n-k -Each row is called a coset. 4.Hamming codes For any positive integer m23.there exists a Hamming code with the following parameters: code length:n=2-1 dimension:k=2-m-1 Number of parity-check symbols:n-k=m Error correcting capability:=l Minimum distance:dmin=3 Note:A field is a set of elements (or symbols)in which we can do addition,subtraction, multiplication,and division without leaving the set.Addition and multiplication satisfy the communicative.associative and distributive laws. Examples:real-number field,complex-number field 1818 3) 估计发送码字c=r-e ˆ . 4) Determine message uˆ from the encoding equation c uG ˆ = ˆ . „ Example: (7, 4) Hamming code. Suppose c = (1001011) is transmitted and r=(1001001) is received. Then 012 ( , , ) (111) T s rH = == sss Let 01 6 e = ( , ,., ) ee e be the error pattern. Since T s eH = , we have the following 3 equations : 1=e0+e3+e5+e6 1=e1+e3+e4+e5 1=e2+e4+e5+e6 There are 16 possible solutions, 其中 e=(0000010)是重量最小，是最可能发生的错误图 样，故c=r e ˆ ⊕ =(10010010)⊕(0000010)=(1001011). „ Standard array c1=0 c2 c3 . cM e2 e2+c2 e2+c3 . e2+cM e3 e3+c2 e3+c3 . e3+cM . . . . . 2 e r 2r + 2 e c 2r + 3 e c . 2r + M e c 2k M = , r = n-k - Each row is called a coset. 4. Hamming codes „ First class of codes devised for error correction. „ For any positive integer m ≥ 3 , there exists a Hamming code with the following parameters: code length: 2 1 m n = − dimension: 2 1 m k m = −− Number of parity-check symbols: n-k=m Error correcting capability: t=1 Minimum distance: dmin=3 Note: A field is a set of elements (or symbols) in which we can do addition, subtraction, multiplication, and division without leaving the set. Addition and multiplication satisfy the communicative, associative and distributive laws. Examples: real-number field, complex-number field