Upload to Canvas a PDF of your work for the following problems.

Assignment: due Monday, 11:59 pm

  1. Use the fact that \(vH^T = \mathbf{0}\) for any codeword \(v\) to prove both parts of the following lemma.

Lemma. Let \(C\) be a systematic linear binary code, and let \(H\) be its parity check matrix.

  1. If \(C\) contains a codeword of weight 1, then \(H\) contains a zero column.
  2. If \(C\) contains a codeword of weight 2, then \(H\) has two identical columns.
  1. The following questions refer to the generating matrix G and message mess given below.
    1. The matrix G is a generating matrix for a systematic linear binary code \(C\). How many codewords are there in this code?
    2. Generate all the codewords in \(C\), and find \(d(C)\) by finding the weight of the lightest nonzero codeword in \(C\).
    3. How many errors can this code correct?
    4. How many different syndromes are there?
    5. Compute the parity check matrix \(H\).
    6. Bob receives the message mess. Find its syndrome.
    7. Determine the coset leader of the coset of mess, and decode the message.
G <- matrix(c(1,0,0,0,0,1,1,1,0,1,1,0,0,1,0,
              0,1,0,0,0,0,1,1,1,0,1,1,0,0,1,
              0,0,1,0,0,1,1,0,1,0,1,1,1,1,0,
              0,0,0,1,0,0,1,1,0,1,0,1,1,1,1,
              0,0,0,0,1,1,1,0,1,1,0,0,1,0,1),
            byrow=TRUE, nrow=5)
mess <- c(0,0,1,0,1,1,1,1,1,1,0,0,0,0,1)