View File - University of Engineering and Technology, Taxila

SolutionComparing (i) and (ii) with the message X in Example 6.2 shows that receivedmessage (i) has one error while (ii) has three errors inserted. So theirsyndromes are2 31 1 11 1 01 0 1ðiÞ S ¼½1011011Š0 1 1¼ 010ð6:20Þ6 1 0 074 0 1 050 0 1An error has been detected since S 6¼ ½0Š, indicating that the codeword isinvalid.2 31 1 11 1 01 0 1ðiiÞ S ¼½1110100Š0 1 1¼ 000ð6:21Þ6 1 0 074 0 1 050 0 1The syndrome is zero. The errors are not detected, because this is a legitimatecodeword in the (7, 4) set. This demonstrates that the (7, 4) Hamming code canalways reliably detect up to two errors: it also leads to the formulation of usefulrules that tell us how many errors a given code can detect or correct. Theserules define the capabilities of linear block codes in terms of weight, W H ,distance, d H , and minimum distance, d H;min .Rule 1: The minimum distance (or Hamming distance) d H;min of a linearblock code is the minimum weight (or Hamming weight), w, of any nonzerocodewords, concisely written asd H;min ¼ min wy2Y H ðY ÞY 6¼0ð6:22ÞThe weight (or Hamming weight) of a code vector Y is the number of nonzerocomponents of Y. The distance (or Hamming distance) between two codevectors Y 1 and Y 2 is the number of components by which they differ. Table6.1 is used as an example of how the weights and distances are calculated. ItCopyright © 2002 by Marcel Dekker, Inc. All Rights Reserved.