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Parity bit
Task number: 4031
Let C be a code of length n with an odd distance d above the alphabet {0,1}. We extend each code word x by one symbol xn+1, determining the parity of the number of ones in x. In particular, xn+1=0 if and only if the first n symbols x contain an even number of ones.
Determine the parameters of the resulting code.
Solution
If C has parameters (n,k,d)2, then the new code has the same size, but the length has increased by 1, but also the distance:
If dist(x,y)=d, then the parity of the number of ones is different in x and y, so we extend each with a different symbol.
Answer
The parameters of the new code are (n+1,k,d+1)2.