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Math Help - generator matrix

  1. #1
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    generator matrix

    Hi is there any one who knows about with generator matrix that know how to solve a problem like this " binary code with the generator matrix G = {1100, 0111, 1010}. Write down the full codebook C". I want to know how do u get the correct answer. I know there needs to be 8 codes words in this , but how to get them is the issue? is it by XOR ing the matrix rows?
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  2. #2
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    Hi Gayan,

    By definition, the rows of the generator matrix form a basis for the linear code.
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  3. #3
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    yes , that his true but how you get them. I got the negation values of the G = {1100, 0111, 1010}.

    which will give me 3 code words. {0011, 1000,0101}. then i XOR the first 3 FROM THE G ,XOR the first 2 I got 1011 and XOR 2nd and 3rd i got 1101.

    as for the inverted matrix from G {0011, 1000,0101} I XOR the first 2 >> 1011 then XOR the 2nd one and 3rd one I got 1101 so all to gether i got

    {1100,0111, 1010,0011,1000,0101,1011,1101} it will make the all code words , but wt if I XOR G = {1100, 0111, 1010} and {0011, 1000,0101}. I will get some different codes. which once should i chose.
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  4. #4
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    Hi Gayan,

    By forming linear combinations over Z_2, a basis with 3 vectors generates (a subspace consisting of) 2^3 = 8 vectors.

    In practise, for a linear code beginning with a basis of 3 codewords, this just involves adding them until you get 8 unique codewords.

    From the basis {1100, 0111, 1010}, we will get, in addition, 0000 (add any codeword to itself), 1100+0111=1011 (sum of the first two in the basis), 0111+1010 = 1101 (sum of the last two in the basis), and the last one comes from adding these previous two we just generated.
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  5. #5
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    Thank you very much i was so confused with this.
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  6. #6
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    Hi

    There is something wrong here. We need to have 8 code words. But here you were giving 7.
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  7. #7
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    2

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  8. #8
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    There is eight. Did you forget 0000?
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