Fall03-quiz2

# Fall03-quiz2 - University of California Berkeley College of...

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1 University of California, Berkeley College of Engineering Computer Science Division EECS Spring 2003 John Kubiatowicz Midterm II December 1, 2003 CS252 Graduate Computer Architecture Your Name: SID Number: Problem Possible Score 1 25 2 25 3 20 4 30 Total 100

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3 Problem #1: Checking the bits The error correction coding process can be viewed as a transformation from one space of bits (the unencoded data) to another (the coded data). 1a) For linear block codes, there are two matrices, the Generator ( G ) and Parity Check ( H ). What are they, how big are they (in terms parameters n and k ), and how are they used? 1b) Define the minimum Hamming distance, d min , of an error correction code (make sure that this definition makes sense for codes like Reed-Solomon of RAID-5 that have more than one bit per symbol): 1c) Name a constraint on the columns of the parity check matrix ( H ) which would indicate that a code had minimum distance d min . 1d) For a code with d min minimum distance, what is the formula for the maximum number of errors, E detect , that can be reliably detected? Why might you be able to detect more than the E detect errors?

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4 1e) For a code with d min minimum distance, what is the formula for the maximum number of random errors, E correct , that can be corrected reliably? 1f) How does your answer to (1e) change if you have knowledge of which symbols are bad (i.e. which symbols have been erased ). How might erased symbols be detected? 1g) What is a systematic error-correction code? Why is this desirable? 1h) Suppose we start with a non-systematic, linear code described by generator G with distance d min . How can we produce a systematic code with generator G’ from G that still has distance d min ? ( hint: what happens if you subtract two code words from the original code? Now think of this operation on G ).
5 1i) Suppose you are willing to use 5 parity bits. What is the maximum number of data bits ( k ) that you can protect with a Hamming code and still get a distance 4 code? What are the matrices G and H for this code? Prove that your code has distance 4. 1j) Explain the function that a circuit that detected and corrected errors in the above code would have to perform. You should tell how to (1) get an “Error detected” signal, E detect (2) get a double-error signal, E double , and (3) a bad bit identifier, E bad .

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