ps4 - CS 151 Complexity Theory Spring 2011 Problem Set 4...

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Unformatted text preview: CS 151 Complexity Theory Spring 2011 Problem Set 4 Out: April 21 Due: April 28 Reminder: you are encouraged to work in groups of two or three; however you must turn in your own write-up and note with whom you worked. You may consult the course notes and the text (Papadimitriou). Please attempt all problems. To facilitate grading, please turn in each problem on a separate sheet of paper and put your name on each sheet. Do not staple the separate sheets. Problem 4 is optional for extra credit. 1. Define ] ZPP to be the class of all languages decided by a probabilistic Turing Machine running in expected polynomial time. That is, for every language L ∈ ] ZPP there is a probabilistic Turing Machine M (with two read-only tapes — the first tape containing the input, and the second tape containing a random bit in every tape square) with the following behavior: on input x ∈ L , M always accepts, on input x ̸∈ L , M always rejects, and for every input x , E[# steps before M halts] = | x | O (1) . Show that ] ZPP = ZPP . 2. List-decoding of the binary Hadamard code. Throughout this problem F 2 is the field with 2 elements (addition and multiplication are performed modulo 2). Given a k-bit message m , the associated Hadamard codeword C ( m ) is described by first producing a linear multivariate polynomial p m ( x ,x 1 ,...,x k − 1 ) = ∑ k − 1 i =0 m i x i , and then evaluating that polynomial at all vectors in the space F k 2 : C ( m ) = ( p m ( w )) w ∈ F k 2 . Thus the codeword has n = 2 k bits, and the w-th bit is the inner product mod 2 of the k-bit vectors m and w . The bits of a codeword C = C ( m ) are naturally indexed by F k 2 ; we write C w (with w ∈ F k 2 ) to mean the w-th coordinate, which is just p m ( w ). Since the distance of the Hadamard code is (1 / 2) n (by Schwartz-Zippel), unique decoding is only possible from up to (1 / 4) n errors. In this problem you will show that eﬃcient list- decoding is possible from a received word R that has suffered up to (1 / 2 − ϵ ) n errors....
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ps4 - CS 151 Complexity Theory Spring 2011 Problem Set 4...

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