6.5 Error-Detecting and Error-Correcting Capability

6.5.1 Weight and Distance of Binary Vectors

It should be clear that not all error patterns can be correctly decoded. The error-correction capability of a code will be investigated by first defining its structure. The Hamming weight w(U) of a codeword U is defined to be the number of nonzero elements in U. For a binary vector, this is equivalent to the number of ones in the vector. For example, if U = 1 0 0 1 0 1 1 0 1, then w(U) = 5. The Hamming distance between two codewords U and V, denoted d(U, V), is defined to be the number of elements in which they differ—for example,

U

= 1 0 0 1 0 1 1 0 1

V

= 0 1 1 1 1 0 1 0 0

d(U, V)

= 6

By the properties of modulo-2 addition, we note ...

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