I’ve recently been revisiting older mathematical papers with the help of AI, checking proofs, correcting errors, and looking for conclusions that were not apparent when the papers were written. Years ago I worked on symmetric objects in the theory of error-correcting codes and had the opportunity to teach coding theory at the Naval Academy. But I fell in love with a different topic, symmetric objects in graph theory, and stopped thinking about codes as an object of research. I’m learning to fall back in love those old objects again.

For quadratic residue codes, the relationship with hyperelliptic curves gives a good example of how one topic can connect with a totally different one – the unity of mathematics. The connection comes from curves of the form

where is a nonempty subset of the finite field . Remarkably, the number of rational points on these curves is closely related to the weights of codewords in certain binary linear codes, called quasi-quadratic residue (QQR) codes. This means that a problem about constructing good codes can be translated into a problem about counting solutions to polynomial equations.

About 20 years ago I wrote a paper, “On quadratic residue codes and hyperelliptic curves” (published in Discrete Mathematics & Theoretical Computer Science 10 (2008), pdf here) to dive into that topic more. I had more questions that answers. Now, 20 years later, I’d like to know what AI thinks of the questions and conjectures in that paper. The result is the AI-aided updated and revised version (2026 pdf here), all my questions were answered by Claude but the conjectures remain open.