4023f11ps7a

4023f11ps7a - Homework #7 Solutions Due: October 26, 2011...

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Homework #7 Solutions Due: October 26, 2011 1. Let C ( F 2 ) 9 be the subset C = ' abcabc a b c j a; b; c 2 F 2 ; where x is 1 if x is 0 and x is 0 if x is 1. Then C is a binary code of length 9. (a) List all of the codewords in C . I Solution. The set of codewords is obtained by letting a , b , c vary over F 2 , that is, a , b , c can be assigned either 0 or 1. Thus, C = f 000000111 ; 001001110 ; 010010101 ; 011011100 ; 100100011 ; 101101010 ; 110110001 ; 111111000 g : J (b) Show that C is not a linear code. I Solution. A linear code must contain the 0 vector, but 000000000 is not in C . Alternatively, one can observe that 000000111+001001110 = 001001001 which is not in C , so C is not closed under addition. J (c) How many errors does C detect and how many errors does it correct. I Solution. It is necessary to compute the minimum distance d ( C ) for the code. Since the code is not linear, it is necessary to compute the distance between any two code words. But codewords in this code are determined by the flrst 3 digits, the second 3 digits repeat the flrst 3, and the third 3 digits are the opposites of the flrst 3. Thus, for each digit that two codewords difier in the flrst 3 digits, this difierence is replicated in the second and third blocks of 3 digits. Hence, the minimum distance must be 3, which corresponds to 1 difierence in the flrst block of 3 digits. Thus, d = d ( C ) = 3, so this code detects d ¡ 1 = 2 errors and corrects b d ¡ 1 2 c = 1 error. J
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4023f11ps7a - Homework #7 Solutions Due: October 26, 2011...

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