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Sixteen-bit messages are transmitted using a Hamming code. How many check bits
are needed to ensure that the receiver can detect and correct single-bit errors? Show
the bit pattern transmitted for the message 1101001100110101. Assume that even parity
is used in the Hamming code.
in Computer Networks by Boss (10.5k points)
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with $m$ parity bits, it can cover bits from $1$ up to $2^m - 1$. If we subtract out the parity bits, we are left with $2^m - m - 1$ bits which are used as data bits.

Parity Bits Total Bits Data Bits
$m$ $2^m - 1$ $ k = 2^m - m - 1$

 

m parity bits can cover bits from $1$ up to $2^m - 1$.

$k = 16$

$16 = 2^m - m - 1$

with $m = 5$, 31 bits can be covered but with 4 parity bits only 15 bits can be covered.

So with 5 check bits receiver can detect and correct single-bit errors.

message = 1101 0011 0011 0101

message + check bits = p1 p2 1 p3 1 0 1 p4 0 0 1 1 0 0 1 p5 1 0 1 0 1

https://gateoverflow.in/239593/hamming-code

 

by Boss (35.4k points)

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