[1101000 0110100 G= 1010010 (4.49) 1110001 It can be verified that the WEF and IRWEF for the (7,4)Hamming code above are, respectively, AX)=1+7X3+7X4+X7,and 4AW,Z)=1+W(3Z2+Z)+W2(3Z+3Z2)+W3(1+3Z)+W*Z In performance analysis it is often useful to group the terms in the IRWEF according to the weights of information words.The following conditional weight enumerating function enumerates the parity block weights for codewords corresponding to the input words of weight w: (4.50) Clearly,A(Z)and IRWEF are related by Aw,Z☑=∑WA(2 4②-” w Weight Enumerators and Performance Bounds Both A(Z)and IRWEF can be used in conjunction with the union bound to compute an upper bound on the bit error probability for ML decoding of the code.Suppose that a length-N codeword x has a weight-w information block of length K and a weight-=parity block.The selection of x by the decoder,as opposed to the correct all-zero codeword,will cause w errors out of a total of K information bits.From (4.45),we have R2长艺Bw+: (4.51) Substitutingd=w+=,(4.51)can be written as R2装B=28R (4.52) ad. where 4404-40 1101000 0110100 1010010 1110001        G (4.49) It can be verified that the WEF and IRWEF for the (7, 4) Hamming code above are, respectively, 3 47 AX X X X ()17 7    , and 2 3 2 2 3 43 AW Z W Z Z W Z Z W Z W Z ( , ) 1 (3 ) (3 3 ) (1 3 )        In performance analysis it is often useful to group the terms in the IRWEF according to the weights of information words. The following conditional weight enumerating function enumerates the parity block weights for codewords corresponding to the input words of weight w:     N K z z Aw Z Aw zZ 0 , ( (4.50) ) Clearly, Aw(Z) and IRWEF are related by   K w w w A W Z W A Z 0 ( , ) ( ) 0 ( , ) ! 1 ( )      w W w w W A W Z w A Z  Weight Enumerators and Performance Bounds Both Aw(Z) and IRWEF can be used in conjunction with the union bound to compute an upper bound on the bit error probability for ML decoding of the code. Suppose that a length-N codeword x has a weight-w information block of length K and a weight-z parity block. The selection of x by the decoder, as opposed to the correct all-zero codeword, will cause w errors out of a total of K information bits. From (4.45), we have , 2 1 0 ( ) K NK b wz w z w P A Pw z K        (4.51) Substituting d  w  z , (4.51) can be written as min , 2 1 ( ) N d b wd w dd w w P A Pd K       min 2 ( ) N d d d B P d    (4.52) where