In a binary Hamming Code the number of check digits is r then number of…
2015
In a binary Hamming Code the number of check digits is r then number of message digits is equal to :
- A.
2𝑟−1
- B.
2𝑟−𝑟−1
- C.
2𝑟−𝑟+1
- D.
2𝑟+𝑟−1
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Correct answer: B
Answer: The number of message digits is 2^r − r − 1.
Derivation:
A Hamming code with r check bits produces 2^r distinct syndromes (binary patterns of the r check bits).
To detect or correct a single-bit error we must distinguish n+1 possibilities: 'no error' or an error in any one of the n bit positions.
Therefore 2^r ≥ n + 1, which implies n ≤ 2^r − 1. Taking the maximum usable n gives n = 2^r − 1.
The number of message (data) bits k = n − r = (2^r − 1) − r = 2^r − r − 1.