Parity Check Detects Single Error

Theorem

Let $\LL$ be a linear $\tuple {n, 2}$-code with a parity check.

Then $\LL$ is such that one transmission error will be detected.


Proof

Let $w$ be a transmitted codeword from $\LL$.

Let $s$ be the sum of the bits of $w$.

Let $w'$ be the received word corresponding to $w$.

Let $s'$ be the sum of the bits of $w'$.


Suppose that $w'$ has one transmission error.

Then either $w'$ has a $1$ instead of $0$, or $0$ instead of $1$.

In the first case, $s' = s + 1$.

In the second case, $s' = s - 1$.

In both cases, the parity of $s'$ is different from the parity of $s$.

That means the parity of $s'$ does not match the parity of the codewords of $\LL$.

Hence the result.

$\blacksquare$


Sources

  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): check digit (checksum)
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