Unformatted text preview: inear code, because
101 + 011 = 110 is not a code word. But if we add 110 to the set, we get a linear code SECTION 6.4. LINEAR BLOCK CODES AND PARITY CALCULATIONS 9 because the sum of any two code words is another code word. The code 000, 101, 011, 110
has a minimum Hamming distance of 2 (that is, the smallest Hamming distance between
any two code words in 2), and can be used to detect all singlebit errors that occur during
the transmission of a code word. You can also verify that the minimum Hamming distance
of this code is equal to the smallest number of 1’s in a nonzero code word. In fact, that’s a
general property of all linear block codes, which we state formally below.
Theorem 6.5 Deﬁne the weight of a code word as the number of 1’s in the word. Then, the minimum Hamming distance of a linear block code is equal to the weight of the nonzero code word with
the smallest weight.
To see why, use the property that the sum of any two code words must also be a code
word, and that the Hamming distance between any two code words is equal to the weight
of their sum (i.e., weight(u + v ) = HD(u, v )). We leave the complete proof of this theorem
as a useful...
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 Fall '13
 HariBalakrishnan

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