<p>You can do a tradeoff between how many errors you correct <span class="math-container">$n$</span> and how many errors you detect <span class="math-container">$m \geq n$</span>, as long as <span class="math-container">$n + m &lt; d$</span>. Just check if the distance to the closest word is at most <span class="math-container">$n$</span>, correct it if it is, and otherwise say there were too many errors.</p>

CC BY-SA 4.0
2023-10-22T08:02:00.657Z
2023-10-22T08:02:00.657Z
Command Master
<p>A block code <span class="math-container">$C$</span>, with minimum distance <span class="math-container">$d$</span> can be used to:</p>
<ol>
<li>Detect <span class="math-container">$d - 1$</span> errors</li>
<li>Correct <span class="math-container">$\lfloor\frac{d - 1}{2}\rfloor$</span> errors</li>
</ol>
<p>However, the above usually assumes that the number of errors that are introduced is below the correctable / detectable threshold.</p>
<p>Suppose codeword <span class="math-container">$x$</span> is transmitted with <span class="math-container">$k &gt; \lfloor\frac{d - 1}{2}\rfloor$</span> errors to get <span class="math-container">$x'$</span>. <span class="math-container">$C$</span> is able to detect correctly that an error has occurred in <span class="math-container">$x'$</span>, but it will erroneously correct the codeword to some <span class="math-container">$y \neq x$</span>.</p>
<p>Are there any ways to detect erroneous corrections?</p>

<it.information-theory><coding-theory>
Detecting Erroneous Corrections
Coziyu
# Detecting Erroneous Corrections

(Asked by: username_0 on 2023-10-21)

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?

**Tags:** it.information-theory, coding-theory

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## 1 answers

### Answer by username_1 on 2023-10-22. Score: 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.


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