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 single-bit 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 non-zero 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 non-zero 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