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 \( x_{n + 1} \), determining the parity of the number of ones in \( x \). In particular, \( x_{n +1} = 0 \) 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 \( \operatorname{dist} (x, y) = d \), 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 \( (n + 1, k, d + 1)_2 \).
