Posted by: **atri** | January 31, 2011

## Lecture 6: Linear codes

In today’s lecture, we went over finite fields and linear subspaces again. In particular, we were able to see that linear codes can be represented succinctly and we get a polynomial time encoding function for linear codes for free. The relevant notes from Fall 2007 are from Lecture 5.

Next lecture, we will see some more interesting properties of linear codes (including an example of a generator matrix).

As promised, here are some pointers to relevant notes/blog posts:

- David Forney’s lectures notes on finite fields. These notes prove pretty much entirely the two theorems on finite field that I stated in class (and much more). A bit of warning: I have not read the notes in any details but they are highly recommended by someone I trust.
**If you are not familiar with finite fields, I highly recommend that you read these notes.** (We will briefly mention the required finite field stuff in the lectures– in particular, I will talk about non-prime fields and their representations using polynomials– but it will not be enough for you to be really comfortable with finite fields.)
- Here is a proof I wrote up that shows that the dimension of a linear subspace is integral. (The proof actually also outlines the algorithm that computes a basis of the linear subspace.)
- Here another set of excellent notes by Madhu Sudan on finite fields and vector spaces over finite fields.

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