Detecting Erroneous Corrections

Question

A block code $C$, with minimum distance $d$ can be used to:

1. Detect $d - 1$ errors
2. Correct $\lfloor\frac{d - 1}{2}\rfloor$ errors

However, the above usually assumes that the number of errors that are introduced is below the correctable / detectable threshold.

Suppose codeword $x$ is transmitted with $k > \lfloor\frac{d - 1}{2}\rfloor$ errors to get $x'$. $C$ is able to detect correctly that an error has occurred in $x'$, but it will erroneously correct the codeword to some $y \neq x$.

Are there any ways to detect erroneous corrections?

Answer 1

You can do a tradeoff between how many errors you correct $n$ and how many errors you detect $m \geq n$, as long as $n + m < d$. Just check if the distance to the closest word is at most $n$, correct it if it is, and otherwise say there were too many errors.