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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.

Difficulty level: Easy task (using definitions and simple reasoning)
Reasoning task
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