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PROFESSOR: So we're slightly into chapter seven, which is the algebra chapter. We're talking about a number of algebraic objects, starting with integers and groups and fields. Polynomials. Our objective in this chapter is simply to get to finite fields so that you have some sense what they are, how they can be constructed, what their parameters are, how you can operate with them by addition, multiplication, so forth, as you would expect in a field. Subtraction, division. And that's really all we're aiming to do. I'm trying to give you a short course in algebra, really, in two lectures or fewer. And clearly I'm going to miss a lot of things. In particular, I'm not going to cover even everything that's in chapter seven, which itself is a highly compressed introduction to finite fields. I'm trying to do this while remaining faithful to the philosophy of this course and other courses at MIT, which is that you should really prove everything ...

PROFESSOR: We're into chapter 11. I've asked Ashish to hand out today a revision of the previous version of chapter 11, plus chapter 12 on the sum-product algorithm, in case we get to it. I'm giving you a complete copy of the problem seven solutions for both weeks, and to get back on track, we're handing out problem set eight. There probably will only be one more problem set that you hand in. All right. We've been getting to the centerpiece of the second half of the course which is codes on graphs, which is the way we're going to get to capacity-achieving codes by building them on graphs of linear complexity and decoding them on the graphs with iterative decoding. I frame this in the language of behavioral realizations. I say a graph is a realization of a code via a behavior, which is the set of all the possible trajectories on the graph. So this is, again, putting it in system theory language. I want to distinguish between two types of variables. ...