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 SOLUTIONS 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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2 [ This page left for π ] 3.141592653589793238462643383279502884197169399375105820974944
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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? The Generator, G , is a k u n matrix used to produce code words from data words: Let d be a k -bit data vector and C be the resulting n- bit code word. Then C=G d. H is an n u (n-k) matrix used to check the parity of a code world and produce a syndrome indicating where errors are located. Let S be an ( n-k) -bit vector, then S=H C. 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): d min is the minimum number of positions (symbols) of difference between two valid code word. If the symbols are bits, this is the number of different bit positions, but can be number of different disk blocks for something like RAID-5. 1c) Name a constraint on the columns of the parity check matrix ( H ) which would indicate that a code had minimum distance d min . If a code with parity check matrix H has minimum distance d min , then every combination of d min -1 columns of H must be independent. This means that no combination of d min -1 columns can be combined to yield zero. Similarly, no combination of d min -2 columns, d min -3 columns, etc. This also means that every column is unique and non-zero. 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? 1 ± PLQ GHWHFW G ( . You might be able to detect more if the code words are not evenly distributed. However, more simply, you can almost always detect a number of errors that is not evenly divisible by d min.
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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? » ¼ » « ¬ « ± 2 1 PLQ GHWHFW G ( 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? If you know which symbols are bad, then you can often correct up to 1 ± PLQ G errors. Erased symbols might be detected by some other means such as checksums over individual symbols (especially makes sense for large symbols such as disk blocks) or even another intra-symbol-level error code.
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