hw10sol - theoretical maximum(C If the code can always...

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Math 55: Discrete Mathematics, Fall 2008 Homework 10 Solutions 3.8: 2(b) - 4 9 2 10 - 4 - 5 4 0 Or, in arithmetic mod 7, 3 2 2 3 3 2 4 0 * 4(c) 2 0 - 3 - 4 - 1 24 - 7 20 29 2 - 10 4 - 17 - 24 - 3 Or, in arithmetic mod 7, 2 0 4 3 6 3 0 6 1 2 4 4 4 4 4 * [5 pts each part] (A) (a) 1 - ( . 95) 4 . 185 (b) 1 - ( . 95) 7 - 7( . 95) 6 ( . 05) . 044 (B) For p = . 05, 1 - h . 713 is the theoretical maximum data rate. The data rate when 4 message bits are encoded with 7 code bits is 4 / 7 . 571, quite a bit less than the theoretical maximum.
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Unformatted text preview: theoretical maximum. * (C) If the code can always correct 2 errors, then every word within Hamming distance 2 of a codeword C must decode to C . For a 15-bit code, the number of such words is 1 + ³ 15 1 ´ + ³ 15 2 ´ = 121 . Hence the number of codewords is at most b 2 15 / 121 c = 270. To encode m message bits requires 2 m codewords, so 2 m can be at most 2 8 = 256....
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