Application of Drinfeld modular curves in Coding Theory

This is an in-person seminar at BIMSA over lunch, aimed to promote communications in the Number Theory teams at BIMSA and YMSC. Each talk is 45 minutes long and does not focus on research results. Instead, we encourage each speaker to discuss either (1) a basic notion in Number Theory or related fields or (2) applications or computational aspects of Number Theory. People interested in Number Theory are welcome to attend.

Abstract：

By Goppa's construction, good towers yield good linear error-correcting codes. The existence of long linear codes with the relative good parameters above the well-known Gilbert-Varshamov bound discovery by Tsfasman et al, provided a vital link between Ihara's quantity and the realm of coding theory. Good towers that are recursive play important roles in the studies of Ihara's quantity, usually constructed from modules curves. Elkies deduced explicit equations of rank-2 Drinfeld modular curves which coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth. In 2015, Bassa, Beelen, Garcia, and Stichtenoth constructed a celebrated (recursive and good) tower (BBGS-tower for short) of curves and outlined a modular interpretation of the defining equations. Soon after that, Gekeler studied in depth the modular curves coming from sparse Drinfeld modules. In this talk, to establish a link between these existing results, I propose a generalized Elkies' Theorem which tells in detail how to directly describe a modular interpretation of the equations of the BBGS tower.