What is the maximum number of errors that can be detected if the distance is 5?
Understand the Problem
The question is asking for the maximum number of errors that can be detected based on a given error-correcting code distance. The distance here refers to the minimum number of differing bits between two valid codewords, which is crucial in determining the error-detection capability.
Answer
The maximum number of detectable errors is $t = d - 1$.
Answer for screen readers
The maximum number of errors that can be detected is given by the formula $t = d - 1$.
Steps to Solve
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Identify the code distance The first step is to recognize the given code distance, which is noted as $d$. This distance is essential for understanding how many errors can be detected.
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Use the error detection formula For an error-correcting code with distance $d$, the maximum number of errors $t$ that can be detected is given by the formula:
$$ t = d - 1 $$
This means that if the distance is 3, for instance, you can detect up to 2 errors.
- Calculate the maximum errors that can be detected Using the identified distance, substitute it into the formula. For example, if $d = 5$, then:
$$ t = 5 - 1 = 4 $$
Thus, the maximum number of detectable errors is 4.
The maximum number of errors that can be detected is given by the formula $t = d - 1$.
More Information
The ability to detect errors is critical in communications and data integrity. Each additional distance in code increases the robustness of the code, allowing more errors to be detected. The concept of error detection is applied in various technologies, including digital communications, data storage, and network transmission.
Tips
- Confusing the error detection capability with error correction capability.
- Forgetting to subtract one from the distance when calculating the maximum number of detectable errors.
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