Near-linear time decoding of Ta-Shma’s codes via splittable regularity

Abstract

We develop new list decoding algorithms for Tanner codes and distance-amplified codes based on bipartite spectral expanders. We show that proofs exhibiting lower bounds on the minimum distance of these codes can be used as certificates discoverable by relaxations in the Sum-of-Squares (SoS) semidefinite programming hierarchy. Combining these certificates with certain entropic proxies to ensure that the solutions to the relaxations cover the entire list, then leads to algorithms for list decoding several families of codes up to the Johnson bound. We prove the following results:-We show that the LDPC Tanner codes of Sipser-Spielman [IEEE Trans. Inf. Theory 1996] andZémor [IEEE Trans. Inf. Theory 2001] with alphabet size q, block-length n and distance δ, based on an expander graph with degree d, can be list-decoded up to distance J q (δ) − ε in time n O d,q (1/ε 4 ) , where J q (δ) denotes the Johnson bound.-We show that the codes obtained via the expander-based distance amplification procedure of Alon, Edmonds and Luby [FOCS 1995] can be list-decoded close to the Johnson bound using the SoS hierarchy, by reducing the list decoding problem to unique decoding of the base code.In particular, starting from any base code unique-decodable up to distance δ, one can obtain near-MDS codes with rate R and distance 1 − R − ε, list-decodable up to the Johnson bound in time n O ε,δ (1) .-We show that the locally testable codes of Dinur et al. [STOC 2022] with alphabet size q, block-length n and distance δ based on a square Cayley complex with generator sets of size d, can be list-decoded up to distance J q (δ) − ε in time n O d,q (1/ε 4 ) , where J q (δ) denotes the Johnson bound.

List Decoding Random Euclidean Codes and Infinite Constellations

“…For instance, recently Dinur et al [41] showed how double samplers give rise to a generic way of amplifying distance so as to enable efficient list decoding algorithms. Followup works by Alev et al [42], Jeronimo et al [43] and Jeronimo et al [44] equipped Ta-Shma's codes (and their variants) with efficient list decoding and unique decoding algorithms using connections to high-dimensional expanders and the Sum-of-Square hierarchy.…”