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Math Help - Coding Theory Help Needed!

  1. #1
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    Coding Theory Help Needed!

    I have a few questions here that I hope you's can help me with:

    1) Does a binary (8,4,5) code exist?

    2)Does a binary (7,3,5) code exist?

    (n,M,d) - n length of codeword, M size of code, d is the minimum distance.

    3) Determine whteher or not the following subsets are supspaces of (i) Z^4 3 and (ii) [B]Z[B]^4 5 respectively.

    (i) S = {x=x1x2x3x4: x1=X3 & x2=x4}

    (ii) S = {x=x1x2x3x4: x1=1}

    Looking forward to some replys!
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  2. #2
    Senior Member Peritus's Avatar
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    1. one can easily show that such a binary code does not exist using the Griesmer bound.
    2. same here.

    3. I do't entirely ubnderstand what's this : (i) Z^4 3 and (ii) [b]Z[b]^4 5
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  3. #3
    Senior Member
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    I am going to rewrite what I think your 3rd question is in my notation. please let me know if I have changed it at all.

    (i) Determine whether the subset S = {x = (x_1,x_2,x_3,x_4)|x_1 = x_3, x_2 = x_4} is a subspace of \mathbf {Z}^{~4}_3<br />


    (ii) Determine whether the subset S = {x = (1,x_2,x_3,x_4)} is a subspace of \mathbf{Z}^{~4}_5<br />

    For both of these you simply need to check the following properties of a subspace (since we know it is a subset):

    closure under addition
    closure under multiplication by a scalar
    non-empty
    existence of additive inverse
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