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

Assignment: due Wednesday, 11:59 pm

  1. In 2007, a new ISBN-13 system was introduced. The check digit \(a_{13}\) is chosen so that \[ a_1 + 3a_2 + a_3 + 3a_4 + a_5 + 3a_6 + a_7 + 3a_8 + a_9 +3a_{10} + a_{11} + 3a_{12} + a_{13} = 0 \] in \(\mathbb{Z}_{10}\).

Show (by finding an example), that it is possible to switch two adjacent digits without changing the check digit. (Of course, the adjacent digits in your example should not be equal.)

  1. Compute the code rate for the Two-Dimensional Parity Check code discussed in the notes.

  2. Hereโ€™s an \(8 \times 8\) Hadamard matrix. (See pages 397-398.)

\[ B = \begin{bmatrix} 1 & 1 & 1 & -1 & 1 & -1 & -1 & -1 \\ 1 & 1 & -1 & -1 & -1 & 1 & -1 & 1 \\ 1 & -1 & 1 & 1 & -1 & -1 & -1 & 1 \\ 1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 \\ 1 & -1 & -1 & -1 & 1 & -1 & 1 & 1 \\ 1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \end{bmatrix} \] Correct the errors in each of the following messages, if possible.

  1. \(\begin{bmatrix} 1 & -1 & 1 & 1 & -1 & 1 & 1 & -1 \end{bmatrix}\)
  2. \(\begin{bmatrix} -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 \end{bmatrix}\)