On our most recent episode of My Favorite Theorem, Kevin Knudson and I talked with University of Nebraska mathematician Judy Walker, who works in the field of coding theory. You can listen here or find the audio, transcript, and show notes at kpknudson.com.
Walker’s field, coding theory, is about how to transmit information over noisy channels and correct the errors that arise when we do. So it seemed very appropriate that our podcast itself was an example of transmitting information over a noisy channel. For our podcasts, we use a live video chat. Technology and internet connections are not perfect, but we could talk to each other despite the glitches that crept in. As Walker explained, the very error-correcting codes she studies are used in online communication. I was excited about the fact that the recording of the podcast itself was an example of the math discussed in the podcast.
Walker’s theorem is probably the most difficult theorem we’ve had on the podcast, at least in terms of being able to spell it based on hearing someone say it. It’s called the Tsfasman-Vladut-Zink theorem, and it is related to how efficient error-correcting codes can be. The basic idea is that for a long time, researchers knew about a lower bound for efficient codes, the Gilbert-Varshamov bound, that says that there are codes at least this efficient. For 30 years, no one could find codes more efficient than the Gilbert-Varshamov bound, and they thought perhaps it was also an upper bound. But with the Tsfasman-Vladut-Zink theorem, researchers showed that there were codes that were more efficient than the Gilbert-Varshamov bound and that realized a known upper bound for efficiency.
Of course, this big picture description hides a lot of the devilish details, like how exactly to measure efficiency. To learn more about those details and the ins and outs of error-correcting codes—and to find out why lemon zest is a perfect pairing for the Tsfasman-Vladut-Zink theorem—listen to or read the full episode here.