Consider a codeset with 4 codewords1 1 1 1 1 1 1 1 0 0 0 0 0

Consider a codeset with 4 codewords:[1 1 1 1 1 1 1 1], [0 0 0 0 0 0 0 0], [1 1 1 1 0 0 0 0], [0 0 0 0 1 1 1 1]

• (1.5 points) What is the Hamming distance between the 1st and 2nd codewords?

• (1.5 points) What is the minimum Hamming distance for this code set?

• (1.5 points) How many bit errors will this code set be able to detect?

Solution

Given code words
   [1 1 1 1 1 1 1 1],
   [0 0 0 0 0 0 0 0],
   [1 1 1 1 0 0 0 0],
   [0 0 0 0 1 1 1 1]
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• (1.5 points) What is the Hamming distance between the 1st and 2nd codewords?

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Answer:
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   1st codeword: [1 1 1 1 1 1 1 1]
   2nd codeword:[0 0 0 0 0 0 0 0]

   Hamming distance is the number of bits 1st and 2nd codeword differs.
  
   Hamming distance = 8
   Since there are 8 bits differs from 1st and 2nd code word
  
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• (1.5 points) What is the minimum Hamming distance for this code set?
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Answer:
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                      [1 1 1 1 1 1 1 1]    [0 0 0 0 0 0 0 0] [1 1 1 1 0 0 0 0] [0 0 0 0 1 1 1 1]
                       -----------------------------------------------------------------------------

   [1 1 1 1 1 1 1 1]           0                   8               4               4
   [0 0 0 0 0 0 0 0]           8                   0               4               4
   [1 1 1 1 0 0 0 0]            4                   4               0               8
   [0 0 0 0 1 1 1 1]           4                   4               8               4
  
   Minimum hamming distance is 0
  

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• (1.5 points) How many bit errors will this code set be able to detect?

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Answer:
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   1st codeword: [1 1 1 1 1 1 1 1]
   2nd codeword:[0 0 0 0 0 0 0 0]
   will detect 8 bit errors.

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