GATE EC 2021 Set 1 — Question 44
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Communications → Error Control Coding → Hamming Distance & Codes
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Question
A digital transmission system uses a systematic linear Hamming code for transmitting data over a noisy channel. If three of the message-codeword pairs in this code , where is the codeword corresponding to the message , are known to be , and , then which of the following is a valid codeword in this code?
Correct answer
(C) 0001011
Solution
In a linear code, any linear combination of codewords is also a codeword. From the given pairs, we can find the codewords for the basis vectors of the message space. Let denote the codeword for message .
Assuming the systematic form is , we find the parity equations:
Testing option C: .
Codeword is , which matches option C.
Assuming the systematic form is , we find the parity equations:
Testing option C: .
Codeword is , which matches option C.
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