arXiv subjects
Publications and source records attributed to Nikhil Shagrithaya.
In this paper we present two reductions between variants of the Code Equivalence problem. We give polynomial-time Karp reductions from Permutation Code Equivalence (PCE) to both Linear Code Equivalence (LCE) and Signed Permutation Code Equivalence (SPCE). Along with a Karp reduction from SPCE to the Lattice Isomorphism Problem (LIP) proved in a paper by Bennett and Win (2024), our second result implies a reduction from PCE to LIP.
We present a general framework for derandomizing random linear codes with respect to a broad class of properties, known as local properties, which encompass several standard notions such as distance, list-decoding, list-recovery, and perfect hashing. Our approach extends the classical Alon-Edmonds-Luby (AEL) construction through a modified formalism of local coordinate-wise linear (LCL) properties, introduced by Levi, Mosheiff, and Shagrithaya (2025). The main theorem demonstrates that if random linear codes satisfy the complement of an LCL property $\mathcal{P}$ with high probability, then one can construct explicit codes satisfying the complement of $\mathcal{P}$ as well, with an enlarged yet constant alphabet size. This gives the first explicit constructions for list recovery, as well as special cases (e.g., list recovery with erasures, zero-error list recovery, perfect hash matrices), with parameters matching those of random linear codes. More broadly, our constructions realize the full range of parameters associated with these properties at the same level of optimality as in the random setting, thereby offering a systematic pathway from probabilistic guarantees to explicit codes that attain them. Furthermore, our derandomization of random linear codes also admits efficient (list) decoding via recently developed expander-based decoders.
We establish an equivalence between two important random ensembles of linear codes: random linear codes (RLCs) and random Reed-Solomon (RS) codes. Specifically, we show that these models exhibit identical behavior with respect to key combinatorial properties -- such as list-decodability and list-recoverability -- when the alphabet size is sufficiently large. We introduce monotone-decreasing local coordinate-wise linear (LCL) properties, a new class of properties tailored for the large alphabet regime. This class encompasses list-decodability, list-recoverability, and their average-weight variants. We develop a framework for analyzing these properties and prove a threshold theorem for RLCs: for any LCL property $\mathcal{P}$, there exists a threshold rate $R_\mathcal{P}$ such that RLCs are likely to satisfy $\mathcal{P}$ when $R < R_\mathcal{P}$ and unlikely to do so when $R > R_\mathcal{P}$. We extend this threshold theorem to random RS codes and show that they share the same threshold $ R_\mathcal{P} $, thereby establishing the equivalence between the two ensembles and enabling a unified analysis of list-recoverability and related properties. Applying our framework, we compute the threshold rate for list-decodability, proving that both random RS codes and RLCs achieve the generalized Singleton bound. This recovers a recent result of Alrabiah, Guruswami, and Li (2023) via elementary methods. Additionally, we prove an upper bound on the list-recoverability threshold and conjecture that this bound is tight. Our approach suggests a plausible pathway for proving this conjecture and thereby pinpointing the list-recoverability parameters of both models. Indeed, following the release of a prior version of this paper, Li and Shagrithaya (2025) used our equivalence theorem to show that random RS codes are near-optimally list-recoverable.
We construct constant-sized ensembles of linear error-correcting codes over any fixed alphabet that can correct a given fraction of adversarial erasures at rates approaching the Singleton bound arbitrarily closely. We provide several applications of our results: 1. Explicit constructions of strong linear seeded symbol-fixing extractors and lossless condensers, over any fixed alphabet, with only a constant seed length and optimal output lengths; 2. A strongly explicit construction of erasure codes on bipartite graphs (more generally, linear codes on matrices of arbitrary dimensions) with optimal rate and erasure-correction trade-offs; 3. A strongly explicit construction of erasure codes on non-bipartite graphs (more generally, linear codes on symmetric square matrices) achieving improved rates; 4. A strongly explicit construction of linear nearly-MDS codes over constant-sized alphabets that can be encoded and decoded in quasi-linear time.
We prove several results on linear codes achieving list-recovery capacity. We show that random linear codes achieve list-recovery capacity with constant output list size (independent of the alphabet size and length). That is, over alphabets of size at least $\ell^{Ω(1/\varepsilon)}$, random linear codes of rate $R$ are $(1-R-\varepsilon, \ell, (\ell/\varepsilon)^{O(\ell/\varepsilon)})$-list-recoverable for all $R\in(0,1)$ and $\ell$. Together with a result of Levi, Mosheiff, and Shagrithaya, this implies that randomly punctured Reed-Solomon codes also achieve list-recovery capacity. We also prove that our output list size is near-optimal among all linear codes: all $(1-R-\varepsilon, \ell, L)$-list-recoverable linear codes must have $L\ge \ell^{Ω(R/\varepsilon)}$. Our simple upper bound combines the Zyablov-Pinsker argument with recent bounds from Kopparty, Ron-Zewi, Saraf, Wootters, and Tamo on the maximum intersection of a "list-recovery ball" and a low-dimensional subspace with large distance. Our lower bound is inspired by a recent lower bound of Chen and Zhang.