Due Wednesday, December 2nd

For Wed:

  • Please know that online FCQs are open.  FCQs are used to evaluate your instructors for reappointment, promotion and tenure, and to inform the department about their teaching effectiveness.  I, personally, greatly appreciate feedback and work to improve my teaching using your feedback.
  • If needed, you’ll want to review the following aspects of linear algebra for studying linear codes:
      • vectors and matrix multiplication
      • vector space and subspace
      • span
      • basis (independence, uniqueness of expression in terms of a basis)
      • dimension
      • kernel of a linear transformation
      • rowspace and column-space
      • rank-nullity theorem
      • transpose of a matrix
  • To Do:
    • Please take some time to fill in this feedback form on aspects of the course.  I list some of the design aspects of our course and ask whether they were helpful or problematic.  It’s a big help to me for designing for future students.  It’s anonymous.
    • Consider the binary code C = {(0,0,1),(1,1,1),(1,0,0),(0,1,0)}.  The alphabet is {0,1}.  This is the same one as last daily post.  Some of the questions that follow are also repeated; you can just revisit your answers.
        1. What is the length of C?
        2. What is d(C) (the minimum distance)?
        3. What are q, n, M and d if C is a q-ary (n,M,d)-code?
        4. How many errors can C detect?
        5. How many errors can C correct?
        6. Show that C is not linear.
        7. Suppose you send the codeword (1,1,1) and 2 errors are made on the noisy channel, in the first and last positions. Explain what message is received and what it decodes to. Was communication successful?
        8. Show that C is a coset of a linear code.  (The definition of a coset is on page 413 of the text, although I talked about it informally in class today:  it’s a translation, by some vector, of a subspace).  What is the linear code?  Call it C’.
        9. Find n and k so that C’ is an [n,k]-code.
        10. Define C’ by linear equation(s).
        11. Give a basis for C’.
        12. Give a matrix for which C’ is the rowspace.