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View File - University of Engineering and Technology, Taxila

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If we consider a message X ¼ð1010Þ, the corresponding codeword Y,generated from (6.4), is231 0 0 0 1 1 10 1 0 0 1 1 0Y ¼ GX ¼ 674 0 0 1 0 1 0 15½1010Š ¼1010010 ð6:18aÞ0 0 0 1 0 1 1The syndrome231 1 1 0 1 0 0S ¼ YH D ¼½1010010Š41 1 0 1 0 1 05 ¼ 000 ð6:18bÞ1 0 1 1 0 0 1If instead <strong>of</strong> the valid received codeword Y, an error is inserted into the secondbit by changing the 0-bit to a 1-bit, <strong>and</strong> the corrupted codeword is written asZ ¼ 1110010ð6:19aÞwhere the underlined bit is the error bit, then the syndrome <strong>of</strong> the newlyreceived codeword is2 31 1 11 1 01 0 1S ¼½1110010Š0 1 1¼ 1106 1 0 074 0 1 050 0 1ð6:19bÞSince S is nonzero, two errors have been detected, indicating that the codewordis incorrect.Example 6.3: If the message word is the same as in Example 6.2 for the (7, 4)Hamming code but received corrupted codewords (i) 1011011 <strong>and</strong> (ii)1110100, determine whether these errors can be detected.Copyright © 2002 by Marcel Dekker, Inc. All Rights Reserved.

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