BARC talk by Rohan Goyal

Tuesday, January 28, 2025, Rohan Goyal, PhD student at MIT, USA, will give a talk on "Fast list-decoding of univariate multiplicity and folded Reed-Solomon codes".

Abstract:
We show that the known list-decoding algorithms for univariate multiplicity and folded Reed-Solomon codes can be made to run in O~(n) time. Univariate multiplicity codes and FRS codes are natural variants of Reed-Solomon codes that were discovered and studied for their applications to list decoding. It is known that for every ϵ>0, and rate r∈(0,1), there exist explicit families of these codes that have rate r and can be list decoded from a (1−r−ϵ) fraction of errors with constant list size in polynomial time (Guruswami & Wang (IEEE Trans. Inform. Theory 2013) and Kopparty, Ron-Zewi, Saraf & Wootters (SIAM J. Comput. 2023)). In this work, we present randomized algorithms that perform the above list-decoding tasks in O~(n), where n is the block-length of the code. Our algorithms have two main components. The first component builds upon the lattice-based approach of Alekhnovich (IEEE Trans. Inf. Theory 2005), who designed a O~(n) time list-decoding algorithm for Reed-Solomon codes approaching the Johnson radius. As part of the second component, we design O~(n) time algorithms for two natural algebraic problems: given a (m+2)-variate polynomial Q(x,y0,…,ym)=Q~(x)+∑mi=0Qi(x)⋅yi the first algorithm solves order-m linear differential equations of the form Q(x,f(x),dfdx,…,dmfdxm)≡0 while the second solves functional equations of the form Q(x,f(x),f(γx),…,f(γmx))≡0, where m is an arbitrary constant and γ is a field element of sufficiently high order. These algorithms can be viewed as generalizations of classical O~(n) time algorithms of Sieveking (Computing 1972) and Kung (Numer. Math. 1974) for computing the modular inverse of a power series, and might be of independent interest.


Bio:
Rohan Goyal is a Ph.D. student at CSAIL, MIT under the supervision of Prof. Yael Kalai. He did his bachelor’s from CMI, India. His research interests span across cryptography, coding theory, and theoretical CS broadly.

Rohan is also a mentor of the Indian mathematical olympiad team and trains high school students for olympiads too.

Host:
Amik Raj Behera