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Question

The parity check matrix of a(7,4) linear block codes is given by,If the code word received at the output of the channel is [1110100]. If there is a possibility for at most one−bit error due to channel noise, then the correct code wordtransmitted at the input of the channel will be.H=⎡⎢⎣111010010110101111001⎤⎥⎦

A
[1 1 1 0 0 0 0]
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B
[1 1 0 0 1 0 0]
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C
[1 0 1 0 1 0 0]
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D
[1 1 1 0 1 0 0]
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Solution

The correct option is D [1 1 1 0 1 0 0]
Syndrome vector, =rHT=[11101000]⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢111101110011100010001⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥=[0 0 0]The syndrome vector has all zeros. It means that, there is error no error in the received code word.So the transmitted code word is same as the received code word.Note: In the problems related to linear block codes, use EXOR opration whenever there is anaddition operation of bits.

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