Explicit Codes approaching Generalized Singleton Bound using Expanders
Fernando Granha Jeronimo, Tushant Mittal, Shashank Srivastava, Madhur Tulsiani
TL;DR
This work presents an explicit construction of codes achieving list-decoding capacity using distance amplification on expander graphs (AEL framework). It generalizes the AEL approach to average-radius list decoding, establishing a local-to-global amplification of the (ε-relaxed) generalized Singleton bound with a constant-alphabet, LDPC-friendly inner/outer-code structure. Central contributions include a detailed SoS-based decoding algorithm that attains radius (k−1)/k(1−R−ε) with list size O(1/ε) for fixed k and ε, along with provable guarantees for inner random-linear and folded-Reed–Solomon inner codes. The approach avoids algebraic interpolation, relying instead on elementary combinatorial and spectral properties of expanders and the Sum-of-Squares hierarchy, yielding explicit LDPC constructions that approach list-decoding capacity. Collectively, the results connect local constraints on expanders to global list-decoding performance, offering practical, implementable codes with strong decoding guarantees near capacity.
Abstract
We construct a new family of explicit codes that are list decodable to capacity and achieve an optimal list size of $O(\frac{1}ε)$. In contrast to existing explicit constructions of codes achieving list decoding capacity, our arguments do not rely on algebraic structure but utilize simple combinatorial properties of expander graphs. Our construction is based on a celebrated distance amplification procedure due to Alon, Edmonds, and Luby [FOCS'95], which transforms any high-rate code into one with near-optimal rate-distance tradeoff. We generalize it to show that the same procedure can be used to transform any high-rate code into one that achieves list decoding capacity. Our proof can be interpreted as a "local-to-global" phenomenon for (a slight strengthening of) the generalized Singleton bound. Using this construction, for every $R, ε\in (0,1)$ and $k \in \mathbb{N}^+$, we obtain an \emph{explicit} family of codes $\mathcal{C} \subseteq Σ^n$, with rate $R$ such that, - They achieve the $ε$-relaxed generalized Singleton bound: for any $g \in Σ^n$ and any list $\mathcal{H}$ of at most $k$ codewords, we have, \[ \underset{h \in \mathcal{H}}{\mathbb{E}} [Δ(g,h)] ~\geq~ \frac{|\mathcal{H}|-1}{|\mathcal{H}|} \cdot (1 - R - ε). \] - The alphabet size is a constant depending only on $ε$ and $k$. - They can be list decoded up to radius $\frac{k-1}{k}(1-R-ε)$, in time $n^{O_{k,ε}(1)}$. As a corollary of our result, we also obtain the first explicit construction of LDPC codes achieving list decoding capacity, and in fact arbitrarily close to the generalized Singleton bound.