Title: On how to try to beat the Gilbert-Varshamov bound
Abstract: The classic Gilbert-Varshamov (GV) bound from the 1950's gives us packings of q-ary Hamming balls in \(q\)-ary Hamming space, with a rate of \(1-H_q(\delta)\) for all \(q < 49\) this is essentially the densest known. In this informal blackboard talk I'll outline a deep rabbit-hole I've gone down the last few months, trying to beat this bound. I'll show a (to my knowledge) heretofore unanalysed class of random codes that achieve the GV bound for a wide range of parameters, and then show that if even one of a certain class of information inequalities holds then these codes can be refined to beat the GV bound. This is work in progress, and is an invitation for folks to come with me further down the rabbit-hole.
Bio: Sidharth (Sid) Jaggi (B.Tech. IIT Bombay 2000, M.S./Ph.D. CalTech 2006, all in electrical engineering, post-doctoral associate MIT 2006): He joined The Chinese University of Hong Kong in 2007, and the School of Mathematics at the University of Bristol in 2020, where he is currently a Professor of Information and Coding Theory. His research group (somewhat unwillingly) calls itself the CAN-DO-IT Team (Codes, Algorithms, Networks: Design and Optimization for Information Theory). Topics he has worked in include sparse recovery/group-testing, covert communication, network coding, and adversarial channels.