Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2311.02263

List Decoding of Tanner and Expander Amplified Codes from Distance Certificates

Fernando Granha Jeronimo·Shashank Srivastava·Madhur Tulsiani
View original source ↗Read source document ↗Export metadata ↓

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: - We show that the LDPC Tanner codes of Sipser-Spielman [IEEE Trans. Inf. Theory 1996] and Z\'{e}mor [IEEE Trans. Inf. Theory 2001] with alphabet size $q$, block-length $n$ and distance $\delta$, based on an expander graph with degree $d$, can be list-decoded up to distance $\mathcal{J}_q(\delta) - \epsilon$ in time $n^{O_{d,q}(1/\epsilon^4)}$, where $\mathcal{J}_q(\delta)$ 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 \emph{any} base code unique-decodable up to distance $\delta$, one can obtain near-MDS codes with rate $R$ and distance $1-R - \epsilon$, list-decodable up to the Johnson bound in time $n^{O_{\epsilon, \delta}(1)}$. - We show that the locally testable codes of Dinur et al. [STOC 2022] with alphabet size $q$, block-length $n$ and distance $\delta$ based on a square Cayley complex with generator sets of size $d$, can be list-decoded up to distance $\mathcal{J}_q(\delta) - \epsilon$ in time $n^{O_{d,q}(1/\epsilon^{4})}$, where $\mathcal{J}_q(\delta)$ denotes the Johnson bound.

Explore related subjects

cs.DScs.ITmath.IT

Keep this discovery

BibTeXRIS

Fernando Granha Jeronimo, Shashank Srivastava, Madhur Tulsiani. 2023-11-03. List Decoding of Tanner and Expander Amplified Codes from Distance Certificates. https://arxiv.org/abs/2311.02263

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗