Probabilistic Guarantees to Explicit Constructions: Local Properties of Linear Codes
Fernando Granha Jeronimo, Nikhil Shagrithaya
TL;DR
The paper develops a general framework to derandomize random linear codes with respect to local properties (distance, list-decoding, list-recovery, and perfect hashing) by extending the Alon–Edmonds–Luby (AEL) construction within a local-coordinate-wise linear (LCL) formalism. It shows that if random linear codes satisfy the complement of an LCL property with high probability, one can explicitly construct codes avoiding the property with parameter regimes matching random codes, at the cost of a constant alphabet blow-up and LDPC structure. The approach relies on a local-to-global transfer using expander graphs, implied local profiles, and robust local-profile descriptions, enabling explicit constructions that replicate random-code parameters for a broad class of properties, including list-recovery and capacity-achieving variants. Efficient decoding is achieved via recent expander-based decoders, making the explicit constructions practically actionable in regimes where decoding is feasible. The results also connect to concurrent work on bounds for random codes and provide corollaries such as explicit perfect-hash matrices and capacity-achieving list-recoverable codes with favorable alphabet sizes.
Abstract
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.