What is the minimum number of check bits required for a Hamming code that protects 4 bits of data?
Understand the Problem
The question is asking for the minimum number of check bits required to generate a Hamming code that can protect 4 bits of data. To solve this, we need to apply the Hamming code formula which relates the number of data bits (k), the number of check bits (r), and the total number of bits (n) in the code, specifically the relationship 2^r >= k + r.
Answer
$r = 3$
Answer for screen readers
The minimum number of check bits required is $r = 3$.
Steps to Solve
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Identify the number of data bits Given in the problem, the number of data bits $k = 4$.
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Set up the inequality We use the Hamming code formula that relates the number of check bits $r$ and data bits $k$. The formula is: $$ 2^r \geq k + r $$ Substituting the value for $k$, we have: $$ 2^r \geq 4 + r $$
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Test values for r We can now test different values of $r$ to find the smallest integer that satisfies the inequality.
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For $r = 1$: $$ 2^1 = 2 \geq 4 + 1 $$ (False)
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For $r = 2$: $$ 2^2 = 4 \geq 4 + 2 $$ (False)
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For $r = 3$: $$ 2^3 = 8 \geq 4 + 3 $$ $$ 8 \geq 7 $$ (True)
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Conclude the minimum check bits The smallest value of $r$ that satisfies the inequality is $r = 3$. Therefore, the minimum number of check bits required is 3.
The minimum number of check bits required is $r = 3$.
More Information
Hamming codes are a form of error-correcting code used in computer science and information theory to detect and correct single-bit errors in data. The formula used here ensures that enough redundancy is included in the code to achieve error correction.
Tips
- Misunderstanding the inequality: Ensure you correctly apply the inequality $2^r \geq k + r$.
- Skipping values for $r$: Test each integer value of $r$ systematically to find the minimum that meets the requirement.
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