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Bit permutations and mappings

  1. Nov 16, 2009 #1
    What is the number of bit permutations of the set {0,1}n and the number of circular right shifts of the set {0,1}n.

    I think the number of bit permuations is 2n, so is there 4 bit permutations here? Namely (0,0), (0,1), (1,0) and (1,1).

    And the right shift is just sending each element one space right isn't it?

    i.e. (0,0) -> (0,0) , (0,1) -> (1,0) etc

    So what does he mean by the number of shifts? 4?

    Find an injective affine linear map (Z/2Z)3 -> (Z/2Z)3 that sends (1,1,1) to (0,0,0).

    In my notes all I have is that

    f(v) = Av + b

    Here I'm trying to find A, with given v = (1,1,1) and f(v) = (0,0,0)

    I also have that for this to be a bijection

    => for (Z/mZ)n -> (Z/mZ)l then n=l and gcd(det(A), m) = 1

    So since this is needed for a bijection am I ok in saying this can be the criteria for it to be injective?

    So if I find an A where (det(A), 2) = 1 , => det(A) can equal 1

    i.e. A =

    0 1 0
    0 0 1
    1 0 0

    and so Av = (1,1,1) and now I can take b = (1,1,1) so that Av+b = (2,2,2) = (0,0,0)?

    Is this all correct? Thanks.
     
  2. jcsd
  3. Nov 17, 2009 #2
    If something doesn't make sense just say and I'll explain what I meant.
     
  4. Nov 17, 2009 #3
    I don't have long until I have to hand this in, would be nice for some input (any at all)!
     
    Last edited: Nov 17, 2009
  5. Nov 18, 2009 #4
    last bump i spose
     
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