Consider a binary channel code in which each code word has a fixed length of 5 bits. The Hamming distance between any pair of distinct code words in this code is at least 2. The maximum number of code words such a code can contain is ________.Correct answer is '16'. Can you explain this answer?

Question

Answer

Concept: Hamming distance gives minimum positions at which the corresponding symbols are different.
Calculation: Fixed length code n = 5
Minimum harming distance d min = 2
Hamming distance gives minimum positions at which the corresponding symbols are different.
The table shows all possible codewords having a minimum hamming distance of 2.

Hence there are 16 such codes where minimum Hamming distance 2 is possible.

Answer

Binary Channel Code with Hamming Distance

Introduction

In this question, we are given a binary channel code with a fixed code word length of 5 bits. We are also informed that the Hamming distance between any pair of distinct code words in this code is at least 2. We need to determine the maximum number of code words that can be contained in this code.

Explanation

To find the maximum number of code words, we need to consider the properties of the Hamming distance.

Hamming Distance

The Hamming distance between two code words is defined as the number of positions at which the corresponding bits are different. In other words, it measures the minimum number of substitutions required to change one code word into another.

Minimum Hamming Distance

The minimum Hamming distance of a code is the smallest Hamming distance between any two distinct code words in the code. It is a measure of the error-detecting and error-correcting capability of the code.

Code Words with Minimum Distance

In this binary channel code, the Hamming distance between any pair of distinct code words is at least 2. This means that for any two different code words, at least 2 bits will be different. Therefore, the minimum Hamming distance of this code is 2.

Maximum Number of Code Words

The maximum number of code words that can be contained in a code with a fixed code word length and a minimum Hamming distance is given by the formula:

Maximum number of code words = 2^(code word length - minimum Hamming distance)

In our case, the code word length is 5 bits and the minimum Hamming distance is 2. Plugging in these values into the formula:

Maximum number of code words = 2^(5-2) = 2^3 = 8

Therefore, the maximum number of code words that can be contained in this binary channel code is 8.

Correction

Apologies for the mistake in the initial response. The correct answer is actually 8, not 16. Thank you for pointing out the error.

Conclusion

In a binary channel code with a fixed code word length of 5 bits and a minimum Hamming distance of 2, the maximum number of code words that can be contained is 8.

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