finalsoln - University of Illinois Fall 2005 ECE 556/CS...

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Unformatted text preview: University of Illinois Fall 2005 ECE 556/CS 577/MATH 579: Solutions to the Final Examination 1 . An ( n, k ) Hamming code over GF( q ) of redundancy r has block length ( q r- 1) / ( q- 1). ( a ) Hence, n = (3 3- 1) / (3- 1) = 13 and k = n- r = 10 . ( b ) Suppose that the predicted outcome of each game is encoded as 0 for a loss, 1 for a tie, and 2 for a win. Each entry can be thought of as a vector in [GF(3)] 13 . Now, the (13 , 10) Hamming code over GF(3) is perfect : the spheres of radius 1 centered on each codeword are disjoint and exhaustive. Put another way, each vector in [GF(3)] 13 is at distance at most 1 from the nearest codeword of the Hamming code. Thus, 3 10 entries (consisting of the codewords of the Hamming code) are necessary and sufficient to guarantee at least a share of second prize. 2 . Let RM ( m, r ) denote the r-th order Reed-Muller code of length 2 m . ( a ) A ( z ; m, , 0) is the weight enumerator of RM ( m, 0) which has two codewords and 1 . Hence, A ( z ; m, , 0) = 1 + z 2 m . The minimum distance of RM ( m, 1) is 2 m- 1 . ( b ) For s, < s < 2 m- 1 , each coset of RM ( m, 0) has two codewords x and x + 1 from RM ( m, 1). Since the two codewords have complementary weights, and both weights must be 2 m- 1 , we conclude that the vectors both are of weight exactly 2 m- 1 and therefore A ( z ; m, , s ) = 2 z 2 m- 1 . From the formula in the problem statement, we get A ( z ; m, 1 , 0) = A ( z ; m, , 0) + (2 m- 1) 2 z 2 m- 1 = 1 + (2 m +1- 2) z 2 m- 1 + z 2 m . ( c ) According to the Plotkin construction, any codeword i RM ( m +1 , r +1) can be expressed uniquely as [ x , x + y ], where x RM ( m, r +1) and y RM ( m, r ). Now suppose that x is a fixed codeword of weight i in the s-th coset of RM ( m, r ). Then, the set of codewords { [ x , x + y ] : y RM ( m, r ) } has weight enumerator z i A ( z ; m, r, s ). By choosing x to be other members of the same coset...
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This note was uploaded on 02/08/2012 for the course ECE 556 taught by Professor Milenkovic,o during the Fall '08 term at University of Illinois, Urbana Champaign.

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finalsoln - University of Illinois Fall 2005 ECE 556/CS...

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