Books on Goppa Codes

Question

During this summer, I am going to study Goppa Codes in Coding Theory. I have relatively good background in Algebra (including Galois Theory, Commutative and Noncommutative Algebra) and I have studied the basics about Codes from Xing's book.

On the other hand, I have never taken a course in Algebraic Geometry and my knowledge in Topology and Projective Geometry is extremely poor.

Therefore, I would kindly ask you the following:

1. Could you please mention some books which are well written, not terse, avoid topology and start any AG notion from the beginning? For example, what would you say about the notes (1), (2), (3) and the relevant chapter from the book "Error-Correction Coding and Decoding" by Tomlinson et al.? Or maybe Pretzel's book "Codes and Algebraic Curves"?
2. Would it be a better idea to study Goppa Codes for first time without using algebraic geometry, like Xing's book?

Moreover please feel free to write more literature for 1. and 2. and write general comments.

Thank you in advance!

Answer 1

A suggestion that occurs to me is

Algebraic function fields and codes by Henning Stichtenoth.

I personally own a copy, and we waded through it in a graduate level study group. Listing a few features. Some may be pros, some cons, depends on how you look at it.

• There is no geometry really. It is all algebra! Instead of a curve its function field is at the focus.
• The necessary geometric results (such as Riemann-Roch) are developed using tools from algebraic number theory alone. That is: (a smooth model of) a curve = a function field, a point = a discrete valuation ring, divisors and R-R are handled in the language of adeles.
• The Galois theory of extensions of function fields is handled in a way that will give you a deja vu, if you have studied algebraic number theory
• Due to all this, he gets to the analogue of Riemann hypothesis as painlessly as possible.

However, I'm not convinced about this approach being best pedagogically. I recall the reactions of the other participants of my study group the day I decided to spend a while with actual curves to explain what divisors of polynomials really look like. Also, I find the way differentials need to be handled in this language a bit off-putting.

Stichtenoth has written another book with the same title. I see this version referred to more often than the older one. I cannot give an account of the differences, if any.

I also discussed this theme with Tom Høholdt, one of the authors of the chapter on AG codes in the Handbook of Coding Theory. He specifically told me that he later set as a goal to teach about Algebraic Geometry codes without Algebraic Geometry. That may be motivated by the fact that his students are engineering majors. IIRC a highlight of that approach was proving R-R by proving that the Feng-Rao decoding algorithm works. He gave me a copy of the manuscript. I would need to do a bit of digging to find when and where it was published.

Anyway, I want to share the sentiment:

General algebraic geometry has (for a reason) a reputation as one of the less accessible parts of mathematics, at least if you are studying it on your own. However, the theory of algebraic curves is much simpler, and expositions exist that prove the powerful results without all the heavy machinery of schemes, sheaf cohomology and such.

If you decide to use Stichtenoth, then you may want to accompany it with books covering basic AG concepts. You will still want to be able to hop between the affine charts of the projective space, when working with curves. I don't know which books to recommend for that purpose. Fulton's book on curves is likely very accessible to you. May be the early chapter in Silverman's Arithmetic of Elliptic Curves? If you end up being the algebra person in a group working on telcomm apps, then someday they may ask you about EC crypto, and you will be able to read Silverman, I'm sure.

Answer 2

There is the English translation of the original Geometry and Codes by Valery D. Goppa, Springer, 1988, written by the man himself.

Also, the book Coding Theory: A first course by San Ling and Chaoping Xing, Cambridge University Press, 2004 has a chapter (Chapter 9) on Goppa codes.