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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.

What is the length of C?

What is d(C) (the minimum distance)?

What are q, n, M and d if C is a q-ary (n,M,d)-code?

How many errors can C detect?

How many errors can C correct?

Show that C is not linear.

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?

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’.