A question about two definitions of a Goppa code

Question

I see two definitions of “Goppa codes”. The first is through polynomials, as in this thesis. This is also true in some of the core textbooks in the field, such as van Lint 3rd edition (1999), and MacWilliams and Sloane (1977).

However, another definition of “Goppa codes” is given on Wikipedia, with more than just polynomials. Goppa codes with this definition are also mixed into the first page of Google Scholar results, like Wirtz 1988 as well as Yang, Kumar, Stichtenoth 1994.

Since these code definitions are both called “Goppa codes”, are they equivalent? And if they’re not equivalent, why are they both named “Goppa codes”?

Answer 1

Answering because the comments don't highlight the historical nuance in naming of Goppa's two distinct classes of codes. In particular, the fact that the Wikipedia page on "Goppa codes" didn't use the modern naming convention correctly until I rewrote it recently [1] shows that there is continued confusion over the naming of his codes.

In 1970 and 1971 Goppa published papers on his first big class of error-correcting codes [2,3], which is the polynomial construction you're talking about. An English description of this was written by Berlekamp in 1973 [4]. These "polynomial" codes are famous, since both his papers won information theory best paper awards [5], and with binary Goppa codes being the core of the McEliece cryptosystem [6,7]. Today, this class of codes are referred to as "Goppa codes". This naming was natural to do since they are important and were constructed by Goppa, so nothing's complicated yet.

Much later though, Goppa developed a second class of codes in 1982, which he called "algebriaco-geometric codes" [8], because they use objects from algebraic geometry (algebraic curves, Riemann-Roch spaces, etc.). These codes were intriguing, but become noteworthy once the paper by Tsfasman, Vladut, and Zink was published soon after in 1982 [9]. The TVZ paper brought attention to these codes because they showed results surpassing the Gilbert-Varshamov bound, which hadn't been pushed in the previous 30 years, a result for which they also won best paper award [5].

However, the naming of this second class of codes fluctuated over time. You can see from reference [9] that they are referred to as "Goppa codes" in the title, since TVZ weren't as heavily versed in coding theory, and might not have known there already existed a class of codes called "Goppa codes". These codes have also been referred to as "algebraic geometric codes" and "geometric Goppa codes" over time. However, in the past decade or so the naming has become more settled in the literature, and today this second class of codes are now referred to as "algebraic geometry codes", or "AG codes" for short.

(Sorry I'm a couple years too late, but I hope this at least helps others in the future.)

References:

1. https://en.wikipedia.org/wiki/Algebraic_geometry_code

2. Goppa, Valery Denisovich. "A new class of linear error-correcting codes." Probl. Inf. Transm. 6 (1970): 300-304.

3. Goppa, Valerii Denisovich. "A rational representation of codes and (L,g)-codes." Problemy Peredachi Informatsii 7.3 (1971): 41-49.

4. Berlekamp, Elwyn. "Goppa codes." IEEE Transactions on Information Theory 19.5 (1973): 590-592.

5. https://www.itsoc.org/honors/information-theory-paper-award

6. https://en.wikipedia.org/wiki/Binary_Goppa_code

7. https://classic.mceliece.org/nist.html

8. Goppa, Valerii Denisovich. "Algebraico-geometric codes." Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 46.4 (1982): 762-781.

9. Tsfasman, Michael A., S. G. Vlădutx, and Th Zink. "Modular curves, Shimura curves, and Goppa codes, better than Varshamov‐Gilbert bound." Mathematische Nachrichten 109.1 (1982): 21-28.