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1 Hamming Distance - Ryerson University

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Linear CodesP. Danziger1 Hamming DistanceThroughout this documentFmeans the binary could define dot product, magnitude and Distance in analogy withRn, but in this case wewould get all vectors having length 0 or 1, not very interesting. Instead we use a different definitionof magnitude and Distance , which is much more useful in this 1 ( Hamming Distance )Given two vectorsu,v Fnwe define thehamming distancebetweenuandv,d(u,v), to be the number of places the Hamming Distance between two vectors is the number of bits we must change to changeone into the the Distance between the vectors 01101010 and differ in four places, so the Hamming distanced(01101010,11011011) = 2 (Weight)Theweightof a vectoru Fnisw(u) =d(u,0), the Distance ofuto thezero weight of a vector is equal to the number of 1 s in it. The weight may be thought of as themagnitude of the the weight of contains 6 ones, sow(11011011) = Error Correcting CodesError correcting codes are used in many places, wherever there is the possibility of errors duringtransmission.

De nition 1 (Hamming distance) Given two vectors u;v 2Fnwe de ne the hamming distance between u and v, d(u;v), to be the number of places where u and v di er. Thus the Hamming distance between two vectors is the number of bits we must change to change one into the other. Example Find the distance between the vectors 01101010 and 11011011. 01101010

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math.ryerson.ca

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www.cpe.ku.ac.th

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