GATE EC 2019 Set 1 — Question 17
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Communications → Error Control Coding → Hamming Distance & Codes
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Question
A linear Hamming code is used to map 4-bit messages to 7-bit codewords. The encoder mapping is linear. If the message 0001 is mapped to the codeword 0000111, and the message 0011 is mapped to the codeword 1100110, then the message 0010 is mapped to
Correct answer
(B) 1100001
Solution
Let the 4-bit message be and the 7-bit codeword be .
Since the encoder mapping is linear, it can be represented by a generator matrix such that .Given mappings:
.Due to linearity, if , then .
Therefore, the codeword for will be .
Performing bitwise XOR:
So, .The final answer is
Since the encoder mapping is linear, it can be represented by a generator matrix such that .Given mappings:
1.Message maps to codeword .
2.Message maps to codeword .
We need to find the codeword for message .Notice that (in binary arithmetic, which is XOR for bits)..Due to linearity, if , then .
Therefore, the codeword for will be .
Performing bitwise XOR:
So, .The final answer is
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