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n0 = 8
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n 1 = 47
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HG =
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1001101 0101011 , 0010111
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HA =
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1 1 0 1
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, HB =
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1100 1010 1111
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Figure 10.13 The conversion matrix P. Source: [NAMB02]. 2002 IEICE Japan.
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Proof Length n1 is less than or equal to the sum of the following three values: the product value of the number of HA matrices in matrix P and the number of columns in HA , the product value of the number of HB matrices in matrix P and the number of columns in HB , and the number of columns in H . The maximum number of columns in HA is equal to that in HB , meaning 2l 2 . The number of HA matrices in P is equal to that of HB matrices in P, meaning np n0 l 1. The maximum number of columns in H is 2l 1 1. Consequently we have the inequality n1 np n0 l 2 2l 1 1: Q.E.D.
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By simply re-arranging the variables, we obtain the inequality (10.9).
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Decoding Procedure The decoding proceduce is demonstrated for the (Bl EC)n0 (SEC)n1 UEP code. The syndromes for the received word V that may include error E are expressed as S V P T HT BEC V0 E PT HT BEC E P T HT , BEC
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H = HBEC P =
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0 0 0 1 0 0 0 0 0 0 1
0 0 0 0 1 0 0 1 0 0 1
0 0 0 0 0 1 0 1 1 0 1
0 0 0 0 0 0 1 1 1 1 1
1 0 0 0 0 0 0 1 1 1 0
0 1 0 0 0 0 0 0 1 1 1
0 0 1 0 0 0 0 1 0 1 0
0 0 0 1 0 0 0 0 1 0 1
0 1 1 0 0 0 0 1 1 0 1
0 1 0 1 0 0 0 0 0 1 0
0 0 1 1 0 0 0 1 1 1 1
0 1 1 1 0 0 0 1 0 0 0
0 1 1 1 1 0 0 0 0 1 1
0 1 1 0 1 0 0 0 1 1 0
0 1 0 1 1 0 0 1 0 0 1
0 1 0 0 1 0 0 1 1 0 0
0 0 1 1 1 0 0 0 1 0 0
0 0 1 0 1 0 0 0 0 0 1
0 0 0 1 1 0 0 1 1 1 0
0 0 0 0 1 0 0 1 0 1 1
0 0 1 1 1 1 0 1 0 0 0
0 0 1 1 0 1 0 0 0 1 1
0 0 1 0 1 1 0 1 1 0 1
0 0 1 0 0 1 0 0 1 1 0
0 0 0 1 1 1 0 0 0 1 0
0 0 0 1 0 1 0 1 0 0 1
0 0 0 0 1 1 0 0 1 1 1
0 0 0 0 0 1 0 1 1 0 0
0 0 0 1 1 1 1 0 1 0 0
0 0 0 1 1 0 1 1 0 0 0
0 0 0 1 0 1 1 1 1 1 1
0 0 0 1 0 0 1 0 0 1 1
0 0 0 0 1 1 1 0 0 0 1
0 0 0 0 1 0 1 1 1 0 1
0 0 0 0 0 1 1 1 0 1 0
0 0 0 0 0 0 1 0 1 1 0
1 0 0 0 1 1 1 0 0 1 0
1 0 0 0 1 1 0 0 1 0 0
1 0 0 0 1 0 1 1 1 1 0
1 0 0 0 1 0 0 1 0 0 0
1 0 0 0 0 1 1 1 0 0 1
1 0 0 0 0 1 0 1 1 1 1
1 0 0 0 0 0 1 0 1 0 1
1 0 0 0 0 0 0 0 0 1 1
1 1 0 0 0 1 1 0 0 0 1
0 1 0 0 0 1 1 0 0 1 0
1 1 0 0 0 1 0 0 1 1 1
0 1 0 0 0 1 0 0 1 0 0
1 1 0 0 0 0 1 1 1 0 1
0 1 0 0 0 0 1 1 1 1 0
1 1 0 0 0 0 0 1 0 1 1
0 1 0 0 0 0 0 1 0 0 0
r =11
n0 = 8
n 1 = 47
Figure 10.14 Parity-check matrix of the (55, 44) (B4EC)8 -(SEC)47 UEP code. Source: [NAMB02]. 2002 IEICE