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# lect01 - 6.895 Essential Coding Theory September 8 2004...

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6.895 Essential Coding Theory September 8, 2004 Lecture 1 Lecturer: Madhu Sudan Scribe: Piotr Mitros 1 Administrative Madhu Sudan To do: Sign up for scribing – everyone must scribe, even listeners. Get added to mailing list Look at problem set 1. Part 1 due in 1 week. 2 Overview of Class Historical overview. The field started by a mathematician, Hamming (1940s-1950). Hamming was looking at magnetic storage devices, where data was stored in 32 bit chunks, but bits would occasionally be ﬂipped. He was investigating how we could do something better. There are several possible solutions: Naive solution: Replicate everything twice. Will correct a one bit error, but for every 3 bits, we only have 1 real bit, so the rate of usage is only 1 . 3 Better solution: Divide everything into 7 bit blocks. In each block, store 4 real bits such that for any one bit ﬂipped, we can figure out which bit was ﬂipped. Hamming did this with the following encoding process: G = 1 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 1 1 Encoding: ( b 1 , b 2 , b 3 , b 4 ) −→ ( b 1 , b 2 , b 3 , b 4 ) G · Here, the multiplication is over F 2 . Claim: If a, b { 0 , 1 } , a = b then a · G and b · G differ in 3 coordinates. This implies that we can correct any one bit error, since with a one bit error, we will be one bit away from the correct string, and at least 2 bits away from the incorrect string. Note we have not proven this claim yet.

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lect01 - 6.895 Essential Coding Theory September 8 2004...

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