An exposition of recent list-size bounds of FRS Codes
Abhibhav Garg, Prahladh Harsha, Mrinal Kumar, Ramprasad Saptharishi, Ashutosh Shankar
TL;DR
The paper surveys recent breakthroughs in bounding the list-size for Folded Reed–Solomon codes near capacity, emphasizing the agreement-graph framework and subspace-restriction analysis. It traces the progression from the KRSW/Tamo bound through Srivastava’s parametric improvements to Chen and Zhang’s ambient-space–independent bounds, culminating in near-optimal $O(1/\varepsilon)$–$O(1/\varepsilon^2)$ list sizes for list-decoding, while showing that these advances do not extend to list-recovery. A central theme is relating list size to the dimension of affine subspaces containing close codewords and to edge counts in agreement graphs, aided by the Guruswami–Kopparty restriction lemma. Despite tighter upper bounds, there remains a fundamental exponential lower bound for list-recovery, underscoring a gap between decoding and recovery capabilities. Overall, the results push Folded Reed–Solomon codes closer to the generalized Singleton bound and clarify the limits of algorithmic list-decoding in this setting, with lingering open questions about efficient realization of these bounds.
Abstract
In the last year, there have been some remarkable improvements in the combinatorial list-size bounds of Folded Reed Solomon codes and multiplicity codes. Starting from the work on Kopparty, Ron-Zewi, Saraf and Wootters (SIAM J. Comput. 2023) (and subsequent simplifications due to Tamo (IEEE Trans. Inform. Theory 2024), we have had dramatic improvements in the list-size bounds of FRS codes due to Srivastava (SODA 2025) and Chen & Zhang (STOC 2025). In this note, we give a short exposition of these three results (Tamo, Srivastava and Chen-Zhang).