Investigating the generalised Hamming distance of some code and information transmission of a binary code.Letting A be an alphabet of size a and S = A n . Meaning the set of all worlds of length n with bits chosen from the alphabet A. We now take w ∈ S.Assume n = 14. How many words in S are of Hamming distance less than or equal to 4 from w?Secondly, assume n = 14 and a = 2 and that w ∈ S is transmitted through a Binary Symmetric Channel with probability p of correct transmission for each individual bit. How would you generalise a formula for the probability that 5 or fewer errors will occur during transmission of w?

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Investigating the generalised Hamming distance of some code and information transmission of a binary code.Letting A be an alphabet of size a and   S  =      A          n      . Meaning the set of all worlds of length n with bits chosen from the alphabet A. We now take   w  ∈  S.Assume   n  =  14. How many words in S are of Hamming distance less than or equal to 4 from w?Secondly, assume   n  =  14 and   a  =  2 and that   w  ∈  S is transmitted through a Binary Symmetric Channel with probability p of correct transmission for each individual bit. How would you generalise a formula for the probability that 5 or fewer errors will occur during transmission of w?

Aliya Moore · Accepted Answer

Step 1To have Hamming distance   ≤  4, we pick those positions that are intended to be different from w and purposely choose them to be different from w at that position, of which we have   a  −  1 choices, hence the desired answer is      ∑          i      =      0        4                      (                    14      i                      )              (  a  −  1      )    i    .Step 2Similarly, for the case where   n  =  14,   a  =  2, and p being the probability of success, we choose the position of mistakes. At those positions, we multiply by probability   1  −  p and at the correct positions, we multiply by p. Hence the expression:      ∑          i      =      0        5                      (                    14      i                      )                  p          14      −      i        (  1  −  p      )    i    .That is your idea was right, just write down 6 terms.

Investigating the generalised Hamming distance of some code and information transmission of a binary code.

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