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Unformatted text preview: SOLUTIONS TO THE MIDTERM FOR MAT 312 Instructions: Please do each of the following 4 problems in the spaces provided. Show some work or give an explanation for each of your answers. (There are two extra sheets of paper at the back of this exam.) In the spaces directly below please print your name and your ID number. Name: ID: 1 2 SOLUTIONS TO THE MIDTERM FOR MAT 312 (1) Let C denote a code containing just the following code words: c 1 =0101010, c 2 =1111111, c 3 =1010101, c 4 =0000000, c 5 =1100001. (a) (worth 6 points) Is C a group code? Solution: The sum of the code words c 1 + c 5 is equal to 1001011, which is not a code word. Thus C is not a group code. (b) (worth 7 points) Compute d for this code. Solution: d is equal to the minimum of all the Hamming dis tances H ( c i , c j ) between different code words. Note that H ( c 1 , c 2 ) = 4, H ( c 1 , c 3 ) = 7, H ( c 1 , c 4 ) = 3, H ( c 1 c 5 ) = 4, H ( c 2 , c 3 ) = 3, H ( c 2 , c 4 ) = 7, H ( c 2 , c 5 ) = 4, H ( c 3 , c 4 ) = 4, H ( c 3 , c 5 ) = 3, H ( c 4 , c 5 ) = 3. Thus d = 3. Remark: Many of you computed d by computing the minimal weight of the nonzero code words, and got the correct answer! I took off points for this because usually this method of computing d only works if...
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 Fall '08
 BESCHER
 Algebra, code words

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