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The syndrome is a single word of length (n k): the number of parity checkbits in the codeword. If the codeword Z is correct, Z ¼ Y ; implying that,andYH D ¼½0ŠS ¼ ZH D ¼½0ŠIf the codeword Z is in error; that is,Z ¼ Y þ eð6:12aÞð6:12bÞð6:13Þthe error in the transmission codeword will be detected if e is not zero, where eis an error vector. SoZ ¼ YH D¼½Y eŠH D ¼ YH D eH DSince YH D from (6.12a) is zero, it follows that error syndromeS ¼ eH Dð6:14Þð6:15ÞIf the syndrome S is nonzero, we know that an error has occurred intransmission of the codeword. Conversely, if the syndrome S is zero, Z is avalid codeword. Of course, S may also be zero for some combinations ofmultiple errors. Efficient decoders use syndrome to represent the error pattern,which can then be corrected. A more elaborate systematic parity check code isthe Hamming code. The Hamming code is an example of error detectionproperties of linear block codes.Example 6.2: Consider a (7, 4) Hamming code. This code has 4 message bitsand 3 parity check bits in a 7-bit codeword. The generator matrix G for thiscode is given by231 0 0 0 1 1 10 1 0 0 1 1 0G ¼ 674 0 0 1 0 1 0 15 ¼½I 40 0 0 1 0 1 1.P 3 Šð6:16ÞUsing (6.10), the parity check matrix can be formed:231 1 1 0 1 0 0H D ¼ 4 1 1 0 1 0 1 05 ¼½P 4 . I 3 Š ð6:17Þ1 0 1 1 0 0 1Copyright © 2002 by Marcel Dekker, Inc. All Rights Reserved.