Structure Theorems (and Fast Algorithms) for List Recovery of Subspace-Design Codes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908699445952512 |
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| author | Goyal, Rohan Guruswami, Venkatesan |
| author_facet | Goyal, Rohan Guruswami, Venkatesan |
| contents | List recovery of error-correcting codes has emerged as a fundamental notion with broad applications across coding theory and theoretical computer science. Folded Reed-Solomon (FRS) and univariate multiplicity codes are explicit constructions which can be efficiently list-recovered up to capacity, namely a fraction of errors approaching $1-R$ where $R$ is the code rate.
Chen and Zhang and related works showed that folded Reed-Solomon codes and linear codes must have list sizes exponential in $1/ε$ for list-recovering from an error-fraction $1-R-ε$. These results suggest that one cannot list-recover FRS codes in time that is also polynomial in $1/ε$. In contrast to such limitations, we show, extending algorithmic advances of Ashvinkumar, Habib, and Srivastava for list decoding, that even if the lists in the case of list-recovery are large, they are highly structured. In particular, we can output a compact description of a set of size only $\ell^{O((\log \ell)/ε)}$ which contains the relevant list, while running in time only polynomial in $1/ε$ (the previously known compact description due to Guruswami and Wang had size $\approx n^{\ell/ε}$). We also improve on the state-of-the-art algorithmic results for the task of list-recovery. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08017 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure Theorems (and Fast Algorithms) for List Recovery of Subspace-Design Codes Goyal, Rohan Guruswami, Venkatesan Information Theory Computational Complexity List recovery of error-correcting codes has emerged as a fundamental notion with broad applications across coding theory and theoretical computer science. Folded Reed-Solomon (FRS) and univariate multiplicity codes are explicit constructions which can be efficiently list-recovered up to capacity, namely a fraction of errors approaching $1-R$ where $R$ is the code rate. Chen and Zhang and related works showed that folded Reed-Solomon codes and linear codes must have list sizes exponential in $1/ε$ for list-recovering from an error-fraction $1-R-ε$. These results suggest that one cannot list-recover FRS codes in time that is also polynomial in $1/ε$. In contrast to such limitations, we show, extending algorithmic advances of Ashvinkumar, Habib, and Srivastava for list decoding, that even if the lists in the case of list-recovery are large, they are highly structured. In particular, we can output a compact description of a set of size only $\ell^{O((\log \ell)/ε)}$ which contains the relevant list, while running in time only polynomial in $1/ε$ (the previously known compact description due to Guruswami and Wang had size $\approx n^{\ell/ε}$). We also improve on the state-of-the-art algorithmic results for the task of list-recovery. |
| title | Structure Theorems (and Fast Algorithms) for List Recovery of Subspace-Design Codes |
| topic | Information Theory Computational Complexity |
| url | https://arxiv.org/abs/2512.08017 |