arXiv subjects

Kuan Cheng

Publications and source records attributed to Kuan Cheng.

At least 19 recordsLinked to original sources

arXiv2026

HAF: Adapting Generalist VLAs to Humanoid Whole-Body Loco-manipulation via Hierarchical Action Flow and Spectral Latent RL

Humanoid robots hold great promise as general-purpose agents in human-centered environments, yet generalist vision-language-action (VLA) foundation models are not readily applicable to humanoid whole-body loco-manipulation. The high dimensionality and interdependence of humanoid motions make it challenging for conventional single-stage VLA architectures to coordinate locomotion, waist posture, and dual-arm manipulation effectively. Moreover, policies trained through offline behavior cloning can remain suboptimal during real-world deployment. Although online reinforcement learning can refine policies through real-world interaction, directly tuning large VLA backbones demands excessive computation and may introduce safety risks during real-robot exploration. To address these bottlenecks, we introduce HAF (Humanoid Adaptation Framework), a two-part framework consisting of HAF-VLA and HAF-Steer that transfers off-the-shelf generalist VLA foundation models to humanoid whole-body loco-manipulation. HAF-VLA is a hierarchical action-flow generator built on a pretrained flow-matching VLA. It splits full-body action denoising into three sequential stages with stage embeddings and cross-stage KV caches that retain kinematic dependencies, avoiding incoherent whole-body actions from one-shot generation. On top of the frozen HAF-VLA, HAF-Steer is a latent offline-to-online RL pipeline that leverages flow-matching invertibility and DCT-based dimensionality reduction to restrict RL optimization to a compact noise subspace and train a regularized SAC policy. This avoids updating the large VLA backbone and enables efficient real-world policy refinement. Evaluated on seven real-world humanoid loco-manipulation tasks, HAF surpasses vanilla single-stage VLA baselines and improves whole-body coordination and task performance. Project website: https://grange007.github.io/HAF .

cs.RO↗

arXiv2026

SEED: Targeted Data Selection by Weighted Independent Set

Data selection seeks to identify a compact yet informative subset from large-scale training corpora, balancing sample quality against collection diversity. We formulate this problem as a Weighted Independent Set (WIS) on a similarity graph, where nodes represent data samples weighted by influence, and edges connect semantically redundant pairs. This formulation naturally yields subsets that are simultaneously high-quality and diverse. However, two challenges arise in practice: naive node weights fail to distinguish informative signals from gradient noise, and edge construction under heterogeneous domain distributions produces structurally imbalanced graphs that bias selection toward sparse regions. To address these issues, we introduce two principled refinements from a unified graph perspective: (1) \textit{node value calibration} that restricts influence estimation to the bilateral salient subspace to ground node importance in task-relevant signals rather than surface-level statistics; (2) \textit{local scale normalization} that adapts edge thresholds to local neighborhood density, mitigating graph imbalance induced by cross-domain distribution shifts. Together, these components yield a robust and scalable data selection pipeline dubbed SEED. We further construct \texttt{Honeybee-Remake-SEED-200K}, a compact multimodal dataset curated by SEED. Extensive experiments show that SEED consistently outperforms state-of-the-art methods on instruction tuning, visual instruction tuning, and semantic segmentation across diverse model families.

cs.LG↗

arXiv2026

When Relaxation Does Not Help: RLDCs with Small Soundness Yield LDCs

Locally decodable codes (LDCs) are error-correcting codes that allow recovery of any single message symbol by probing only a small number of positions from the (possibly corrupted) codeword. Relaxed locally decodable codes (RLDCs) further allow the decoder to output a special failure symbol $\bot$ on a corrupted codeword. While known constructions of RLDCs achieve much better parameters than standard LDCs, it is intriguing to understand the relationship between LDCs and RLDCs. On the one hand, separation results (i.e., the existence of $q$-query RLDCs that are not $q$-query LDCs) are known for $q=3$ (Gur, Minzer, Weissenberg, and Zheng, STOC 2026) and $q \geq 15$ (Grigorescu, Kumar, Manohar, and Mon, STOC 2026). On the other hand, prior work (Block, Blocki, Cheng, Grigorescu, Li, Zheng, and Zhu, CCC 2023) shows that any $2$-query RLDC also gives a $2$-query LDC, and Grigorescu, Kumar, Manohar, and Mon (STOC 2026) show that any \emph{linear} $3$-query RLDC is also a linear $3$-query LDC. Furthermore, Grigorescu, Kumar, Manohar, and Mon (STOC 2026) show that when the soundness error of a \emph{linear} $q$-query RLDC with perfect completeness is below some threshold $s(q)$, the code must also be a linear $q$-query LDC with comparable parameters. In this work, we extend the main result of Grigorescu, Kumar, Manohar, and Mon (STOC 2026) by removing the linearity requirement in the nonadaptive setting. Specifically, we show that every nonadaptive $(q,\delta,1,s)$-RLDC over a finite alphabet $\Sigma$ with $s<|\Sigma|^{-q}$ yields a $q$-query LDC with comparable decoding radius and error. Our results also extend to the setting of locally correctable codes (LCCs) and relaxed locally correctable codes (RLCCs). From this, we also obtain lower bounds for nonadaptive RLDCs from known LDC lower bounds.

cs.IT↗

arXiv2026

Exponential Lower Bounds for 2-query Relaxed Locally Decodable Codes

Locally Decodable Codes (LDCs) are error-correcting codes $C\colon\Sigma^n\rightarrow \Sigma^m,$ encoding \emph{messages} in $\Sigma^n$ to \emph{codewords} in $\Sigma^m$, with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length $m$ that is super-polynomial in $n$, for codes with constant query complexity and constant alphabet size. In a very surprising result, Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (SICOMP 2006) show how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs. In this work, we study RLDCs in the standard Hamming-error setting. We prove an exponential lower bound on the length of Hamming RLDCs making $2$ queries (even adaptively) over the binary alphabet. This answers a question explicitly raised by Gur and Lachish (SICOMP 2021) and is the first exponential lower bound for RLDCs. Combined with the results of Ben-Sasson et al., our result exhibits a ``phase-transition''-type behavior on the codeword length for some constant-query complexity. We achieve these lower bounds via a transformation of RLDCs to standard Hamming LDCs, using a careful analysis of restrictions of message bits that fix codeword bits.

cs.IT↗

arXiv2025

RoboMIND 2.0: A Multimodal, Bimanual Mobile Manipulation Dataset for Generalizable Embodied Intelligence

While data-driven imitation learning has revolutionized robotic manipulation, current approaches remain constrained by the scarcity of large-scale, diverse real-world demonstrations. Consequently, the ability of existing models to generalize across long-horizon bimanual tasks and mobile manipulation in unstructured environments remains limited. To bridge this gap, we present RoboMIND 2.0, a comprehensive real-world dataset comprising over 310K dual-arm manipulation trajectories collected across six distinct robot embodiments and 739 complex tasks. Crucially, to support research in contact-rich and spatially extended tasks, the dataset incorporates 12K tactile-enhanced episodes and 20K mobile manipulation trajectories. Complementing this physical data, we construct high-fidelity digital twins of our real-world environments, releasing an additional 20K-trajectory simulated dataset to facilitate robust sim-to-real transfer. To fully exploit the potential of RoboMIND 2.0, we propose MIND-2 system, a hierarchical dual-system frame-work optimized via offline reinforcement learning. MIND-2 integrates a high-level semantic planner (MIND-2-VLM) to decompose abstract natural language instructions into grounded subgoals, coupled with a low-level Vision-Language-Action executor (MIND-2-VLA), which generates precise, proprioception-aware motor actions.

cs.RO↗

arXiv2025

ChainV: Atomic Visual Hints Make Multimodal Reasoning Shorter and Better

Recent advances in multimodal reasoning models have demonstrated impressive capabilities across text and vision. However, even leading models exhibit redundant self-reflection when generating lengthy reasoning chains. While training-free CoT compression methods have emerged in the LLMs domain, they rely on static visual references and thus provide limited gains for multimodal reasoning. Therefore, we propose ChainV, a framework that dynamically integrates visual hints into the reasoning process, thereby making multimodal reasoning shorter and better. Specifically, ChainV first performs a coarse visual patch selection based on the previous reasoning step, then refines it by identifying the most representative atomic visual hint according to the averaged attention intensity. Additionally, ChainV introduces a consistency-based evaluation mechanism to assess the reliability of the chosen hint, guiding the model to adaptively adjust its level of self-reflection. Eventually, the pixel coordinates of the selected visual hint and its reliability are incorporated into thinking with a Bernoulli stochastic process. Experiments indicate that our method significantly improves reasoning accuracy and efficiency, especially on math-intensive benchmarks where visual hints are crucial for multi-step symbolic reasoning. For example, ChainV achieves $2.3\%$ improvement on the MathVista within MIMO-VL-RL, while reducing inference latency by $51.4\%$ and shortening output token length by $24.5\%$.

cs.CV↗

arXiv2025

AutoV: Loss-Oriented Ranking for Visual Prompt Retrieval in LVLMs

Inspired by text prompts in large language models, visual prompts have been explored to enhance the perceptual capabilities of large vision-language models (LVLMs). However, performance tends to saturate under single visual prompt designs, making further prompt engineering increasingly ineffective. To address this limitation, we shift from prompt engineering to prompt retrieval and propose AutoV, a lightweight framework for instance-adaptive visual prompt identification. Given an input image and a textual query, AutoV automatically locates the most suitable visual prompt from a diverse candidate pool. Training such a retrieval framework requires prompt-level supervision, yet prompt quality is inherently ambiguous and difficult to assess reliably, even for humans. To enable automatic supervision, we evaluate visual prompts using a pre-trained LVLM and label them according to their prediction losses. Using the loss-oriented ranking as a robust training signal, AutoV learns to retrieve the query-aware optimal prompt for each instance without manual annotation. Experiments indicate that AutoV enhances the performance of various LVLMs on image understanding, captioning, grounding, and classification tasks. For example, AutoV improves LLaVA-OV by $\textbf{10.2}\%$ on VizWiz and boosts Qwen2.5-VL by $\textbf{3.8}\%$ on MMMU, respectively.

cs.CV↗

arXiv2025

Weighted Pseudorandom Generators for Read-Once Branching Programs via Weighted Pseudorandom Reductions

We study weighted pseudorandom generators (WPRGs) and derandomizations for read-once branching programs (ROBPs). Denote $n$ and $w$ as the length and the width of a ROBP. We have the following results. For standard ROBPs, we give an explicit $\varepsilon$-WPRG with seed length $$O\left(\frac{\log n\log (nw)}{\max\left\{1,\log\log w-\log\log n\right\}}+\log w \left(\log\log\log w-\log\log\max\left\{2,\frac{\log w}{\log \frac{n}{\varepsilon}}\right\}\right)+\log\frac{1}{\varepsilon}\right).$$ For permutation ROBPs with unbounded widths and single accept nodes, we give an explicit $\varepsilon$-WPRG with seed length $$O\left( \log n\left( \log\log n + \sqrt{\log(1/\varepsilon)} \right)+\log(1/\varepsilon)\right). $$ We also give a new Nisan-Zuckerman style derandomization for regular ROBPs with width $w$, length $n = 2^{O(\sqrt{\log w})}$, and multiple accept nodes. We attain optimal space complexity $O(\log w)$ for arbitrary approximation error $\varepsilon = 1/\text{poly} (w)$. All our results are based on iterative weighted pseudorandom reductions, which can iteratively reduce fooling long ROBPs to fooling short ones.

cs.CC↗

arXiv2024

SparseVLM: Visual Token Sparsification for Efficient Vision-Language Model Inference

In vision-language models (VLMs), visual tokens usually bear a significant amount of computational overhead despite sparsity of information in them when compared to text tokens. To address this, most existing methods learn a network to prune redundant visual tokens using certain training data. Differently, we propose a text-guided training-free token optimization mechanism dubbed SparseVLM that eliminates the need of extra parameters or fine-tuning costs. Given that visual tokens complement text tokens in VLM's linguistic reasoning, we select relevant text tokens to rate the significance of visual tokens using self-attention matrices and, then, prune visual tokens using the proposed strategy to maximize sparsity while retaining information. In particular, we introduce a rank-based strategy to adaptively determine the sparsification ratio for each layer, alongside a token recycling method that compresses pruned tokens into more compact representations. Experimental results show that SparseVLM increases the efficiency of various VLMs in a number of image and video understanding tasks. For example, LLaVA when equipped with SparseVLM achieves 54% reduction in FLOPs, 37% decrease in CUDA latency while maintaining 97% of its original accuracy. Our code is available at https://github.com/Gumpest/SparseVLMs.

cs.CV↗

arXiv2024

Unveiling the Tapestry of Consistency in Large Vision-Language Models

Large vision-language models (LVLMs) have recently achieved rapid progress, exhibiting great perception and reasoning abilities concerning visual information. However, when faced with prompts in different sizes of solution spaces, LVLMs fail to always give consistent answers regarding the same knowledge point. This inconsistency of answers between different solution spaces is prevalent in LVLMs and erodes trust. To this end, we provide a multi-modal benchmark ConBench, to intuitively analyze how LVLMs perform when the solution space of a prompt revolves around a knowledge point. Based on the ConBench tool, we are the first to reveal the tapestry and get the following findings: (1) In the discriminate realm, the larger the solution space of the prompt, the lower the accuracy of the answers. (2) Establish the relationship between the discriminative and generative realms: the accuracy of the discriminative question type exhibits a strong positive correlation with its Consistency with the caption. (3) Compared to open-source models, closed-source models exhibit a pronounced bias advantage in terms of Consistency. Eventually, we ameliorate the consistency of LVLMs by trigger-based diagnostic refinement, indirectly improving the performance of their caption. We hope this paper will accelerate the research community in better evaluating their models and encourage future advancements in the consistency domain. The project is available at https://github.com/foundation-multimodal-models/ConBench.

cs.CV↗

arXiv2023

When can an expander code correct $\Omega(n)$ errors in $O(n)$ time?

Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph $G$ together with a linear inner code $C_0$. Expander codes are Tanner codes whose defining bipartite graph $G$ has good expansion property. This paper is motivated by the following natural and fundamental problem in decoding expander codes: What are the sufficient and necessary conditions that $\delta$ and $d_0$ must satisfy, so that \textit{every} bipartite expander $G$ with vertex expansion ratio $\delta$ and \textit{every} linear inner code $C_0$ with minimum distance $d_0$ together define an expander code that corrects $\Omega(n)$ errors in $O(n)$ time? For $C_0$ being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT'96) showed that $\delta>3/4$ is sufficient; later Viderman (ACM-TOCT'13) improved this to $\delta>2/3-\Omega(1)$ and he also showed that $\delta>1/2$ is necessary. For general linear code $C_0$, the previously best-known result of Dowling and Gao (IEEE-TIT'18) showed that $d_0=\Omega(c\delta^{-2})$ is sufficient, where $c$ is the left-degree of $G$. In this paper, we give a near-optimal solution to the above question for general $C_0$ by showing that $\delta d_0>3$ is sufficient and $\delta d_0>1$ is necessary, thereby also significantly improving Dowling-Gao's result. We present two novel algorithms for decoding expander codes, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius.

cs.IT↗

arXiv2023

FreeKD: Knowledge Distillation via Semantic Frequency Prompt

Knowledge distillation (KD) has been applied to various tasks successfully, and mainstream methods typically boost the student model via spatial imitation losses. However, the consecutive downsamplings induced in the spatial domain of teacher model is a type of corruption, hindering the student from analyzing what specific information needs to be imitated, which results in accuracy degradation. To better understand the underlying pattern of corrupted feature maps, we shift our attention to the frequency domain. During frequency distillation, we encounter a new challenge: the low-frequency bands convey general but minimal context, while the high are more informative but also introduce noise. Not each pixel within the frequency bands contributes equally to the performance. To address the above problem: (1) We propose the Frequency Prompt plugged into the teacher model, absorbing the semantic frequency context during finetuning. (2) During the distillation period, a pixel-wise frequency mask is generated via Frequency Prompt, to localize those pixel of interests (PoIs) in various frequency bands. Additionally, we employ a position-aware relational frequency loss for dense prediction tasks, delivering a high-order spatial enhancement to the student model. We dub our Frequency Knowledge Distillation method as FreeKD, which determines the optimal localization and extent for the frequency distillation. Extensive experiments demonstrate that FreeKD not only outperforms spatial-based distillation methods consistently on dense prediction tasks (e.g., FreeKD brings 3.8 AP gains for RepPoints-R50 on COCO2017 and 4.55 mIoU gains for PSPNet-R18 on Cityscapes), but also conveys more robustness to the student. Notably, we also validate the generalization of our approach on large-scale vision models (e.g., DINO and SAM).

cs.CV↗

arXiv2023

Random Shortening of Linear Codes and Applications

Random linear codes (RLCs) are well known to have nice combinatorial properties and near-optimal parameters in many different settings. However, getting explicit constructions matching the parameters of RLCs is challenging, and RLCs are hard to decode efficiently. This motivated several previous works to study the problem of partially derandomizing RLCs, by applying certain operations to an explicit mother code. Among them, one of the most well studied operations is random puncturing, where a series of works culminated in the work of Guruswami and Mosheiff (FOCS' 22), which showed that a random puncturing of a low-biased code is likely to possess almost all interesting local properties of RLCs. In this work, we provide an in-depth study of another, dual operation of random puncturing, known as random shortening, which can be viewed equivalently as random puncturing on the dual code. Our main results show that for any small $\varepsilon$, by starting from a mother code with certain weaker conditions (e.g., having a large distance) and performing a random (or even pseudorandom) shortening, the new code is $\varepsilon$-biased with high probability. Our results hold for any field size and yield a shortened code with constant rate. This can be viewed as a complement to random puncturing, and together, we can obtain codes with properties like RLCs from weaker initial conditions. Our proofs involve several non-trivial methods of estimating the weight distribution of codewords, which may be of independent interest.

cs.IT↗

arXiv2023

On k-Mer-Based and Maximum Likelihood Estimation Algorithms for Trace Reconstruction

The goal of the trace reconstruction problem is to recover a string $x\in\{0,1\}^n$ given many independent {\em traces} of $x$, where a trace is a subsequence obtained from deleting bits of $x$ independently with some given probability $p\in [0,1).$ A recent result of Chase (STOC 2021) shows how $x$ can be determined (in exponential time) from $\exp(\widetilde{O}(n^{1/5}))$ traces. This is the state-of-the-art result on the sample complexity of trace reconstruction. In this paper we consider two kinds of algorithms for the trace reconstruction problem. Our first, and technically more involved, result shows that any $k$-mer-based algorithm for trace reconstruction must use $\exp(\Omega(n^{1/5}))$ traces, under the assumption that the estimator requires $poly(2^k, 1/\varepsilon)$ traces, thus establishing the optimality of this number of traces. The analysis of this result also shows that the analysis technique used by Chase (STOC 2021) is essentially tight, and hence new techniques are needed in order to improve the worst-case upper bound. Our second, simple, result considers the performance of the Maximum Likelihood Estimator (MLE), which specifically picks the source string that has the maximum likelihood to generate the samples (traces). We show that the MLE algorithm uses a nearly optimal number of traces, \ie, up to a factor of $n$ in the number of samples needed for an optimal algorithm, and show that this factor of $n$ loss may be necessary under general ``model estimation'' settings.

cs.IT↗

arXiv2023

Linear Insertion Deletion Codes in the High-Noise and High-Rate Regimes

This work continues the study of linear error correcting codes against adversarial insertion deletion errors (insdel errors). Previously, the work of Cheng, Guruswami, Haeupler, and Li \cite{CGHL21} showed the existence of asymptotically good linear insdel codes that can correct arbitrarily close to $1$ fraction of errors over some constant size alphabet, or achieve rate arbitrarily close to $1/2$ even over the binary alphabet. As shown in \cite{CGHL21}, these bounds are also the best possible. However, known explicit constructions in \cite{CGHL21}, and subsequent improved constructions by Con, Shpilka, and Tamo \cite{9770830} all fall short of meeting these bounds. Over any constant size alphabet, they can only achieve rate $< 1/8$ or correct $< 1/4$ fraction of errors; over the binary alphabet, they can only achieve rate $< 1/1216$ or correct $< 1/54$ fraction of errors. Apparently, previous techniques face inherent barriers to achieve rate better than $1/4$ or correct more than $1/2$ fraction of errors. In this work we give new constructions of such codes that meet these bounds, namely, asymptotically good linear insdel codes that can correct arbitrarily close to $1$ fraction of errors over some constant size alphabet, and binary asymptotically good linear insdel codes that can achieve rate arbitrarily close to $1/2$.\ All our constructions are efficiently encodable and decodable. Our constructions are based on a novel approach of code concatenation, which embeds the index information implicitly into codewords. This significantly differs from previous techniques and may be of independent interest. Finally, we also prove the existence of linear concatenated insdel codes with parameters that match random linear codes, and propose a conjecture about linear insdel codes.

cs.IT↗

arXiv2022

On The Relative Error of Random Fourier Features for Preserving Kernel Distance

The method of random Fourier features (RFF), proposed in a seminal paper by Rahimi and Recht (NIPS'07), is a powerful technique to find approximate low-dimensional representations of points in (high-dimensional) kernel space, for shift-invariant kernels. While RFF has been analyzed under various notions of error guarantee, the ability to preserve the kernel distance with \emph{relative} error is less understood. We show that for a significant range of kernels, including the well-known Laplacian kernels, RFF cannot approximate the kernel distance with small relative error using low dimensions. We complement this by showing as long as the shift-invariant kernel is analytic, RFF with $\mathrm{poly}(\epsilon^{-1} \log n)$ dimensions achieves $\epsilon$-relative error for pairwise kernel distance of $n$ points, and the dimension bound is improved to $\mathrm{poly}(\epsilon^{-1}\log k)$ for the specific application of kernel $k$-means. Finally, going beyond RFF, we make the first step towards data-oblivious dimension-reduction for general shift-invariant kernels, and we obtain a similar $\mathrm{poly}(\epsilon^{-1} \log n)$ dimension bound for Laplacian kernels. We also validate the dimension-error tradeoff of our methods on simulated datasets, and they demonstrate superior performance compared with other popular methods including random-projection and Nystr\"{o}m methods.

cs.LG↗

arXiv2022

On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors

Locally Decodable Codes (LDCs) are error-correcting codes $C:\Sigma^n\rightarrow \Sigma^m$ with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length $m$ that is super-polynomial in $n$, for codes with constant query complexity and constant alphabet size. In a very surprising result, Ben-Sasson et al. showed how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs. In this work, we study RLDCs in the standard Hamming-error setting, and introduce their variants in the insertion and deletion (Insdel) error setting. Insdel LDCs were first studied by Ostrovsky and Paskin-Cherniavsky, and are further motivated by recent advances in DNA random access bio-technologies, in which the goal is to retrieve individual files from a DNA storage database. Our first result is an exponential lower bound on the length of Hamming RLDCs making 2 queries, over the binary alphabet. This answers a question explicitly raised by Gur and Lachish. Our result exhibits a "phase-transition"-type behavior on the codeword length for constant-query Hamming RLDCs. We further define two variants of RLDCs in the Insdel-error setting, a weak and a strong version. On the one hand, we construct weak Insdel RLDCs with with parameters matching those of the Hamming variants. On the other hand, we prove exponential lower bounds for strong Insdel RLDCs. These results demonstrate that, while these variants are equivalent in the Hamming setting, they are significantly different in the insdel setting. Our results also prove a strict separation between Hamming RLDCs and Insdel RLDCs.

cs.IT↗

arXiv2021

Improved Decoding of Expander Codes

We study the classical expander codes, introduced by Sipser and Spielman \cite{SS96}. Given any constants $0< \alpha, \varepsilon < 1/2$, and an arbitrary bipartite graph with $N$ vertices on the left, $M < N$ vertices on the right, and left degree $D$ such that any left subset $S$ of size at most $\alpha N$ has at least $(1-\varepsilon)|S|D$ neighbors, we show that the corresponding linear code given by parity checks on the right has distance at least roughly $\frac{\alpha N}{2 \varepsilon }$. This is strictly better than the best known previous result of $2(1-\varepsilon ) \alpha N$ \cite{Sudan2000note, Viderman13b} whenever $\varepsilon < 1/2$, and improves the previous result significantly when $\varepsilon $ is small. Furthermore, we show that this distance is tight in general, thus providing a complete characterization of the distance of general expander codes. Next, we provide several efficient decoding algorithms, which vastly improve previous results in terms of the fraction of errors corrected, whenever $\varepsilon < \frac{1}{4}$. Finally, we also give a bound on the list-decoding radius of general expander codes, which beats the classical Johnson bound in certain situations (e.g., when the graph is almost regular and the code has a high rate). Our techniques exploit novel combinatorial properties of bipartite expander graphs. In particular, we establish a new size-expansion tradeoff, which may be of independent interests.

cs.IT↗