arXiv subjects

Songtao Mao

Publications and source records attributed to Songtao Mao.

10 recordsLinked to original sources

arXiv2026

PASCAL: A Phase-Aware Shared-Cache Model for Parallel Scans

In modern AI Accelerators and GPGPUs, many concurrent cores repeatedly access the same shared data. This pattern occurs in attention, where different query tiles share the same K/V block, GEMM, where every tile in a row reads the same panel, and many other operators. We name this pattern parallel scan. Due to a significant amount of data reuse in this pattern, the cache is expected to capture as much data reuse as possible and largely reduce requests sent to the main memory for both performance and energy consumption concerns. However, in reality, because of the intrinsic asynchrony of multi-cores, the actual cache miss rate and DRAM traffic can be much higher compared to ideal cases. In this paper, we propose PASCAL, a shared-cache model for parallel scans. It is aware of the dynamic feature of progress divergence across multi-cores, correlate the divergence with the combination of different factors such as occupancy, and predicts the cache miss rate before execution. Because prediction needs no target trace, timing, or counters, PASCAL supports design-space exploration at scales where cycle-accurate simulation is impractical, and its policy-independent bound states how much traffic no replacement policy can avoid. A MAPE of 13.84% is achieved in a 60-configuration dataset with various software pipeline depths, occupancies, and memory access data paths on an NVIDIA GB10 GPU, against 44.79% for physical-wave TileSight and 54.16% for exact symbolic SDCM.

cs.PF↗

arXiv2026

Abelian Cayley High-Dimensional Expanders with Polylogarithmic Degree

We construct an explicit infinite family of simple two-dimensional Cayley complexes over $\mathbb{F}_2^n$ whose degree is polynomial in $n$ and whose nontrivial vertex-link eigenvalues lie in $[-\lambda,\lambda]$ for every fixed $\lambda>0$. For every fixed $d\ge2$, we also obtain an explicit infinite family of weighted $d$-dimensional Cayley complexes over $\mathbb{F}_2^n$ with codimension-two local spectral norm at most $1/d$ and Cayley degree $\Theta_d(n)$. Our two-dimensional construction uses evaluation at rational points of algebraic curves to produce projective direction sets and many functions affine along these directions, which may be useful for further constructions and improvements.

math.CO↗

arXiv2026

The Polynomial-Time Low-Degree Conjecture is False

The low-degree method and its associated lower bounds are widely used to guide algorithm design and to provide evidence of computational hardness in average-case inference, high-dimensional statistics, random optimization, and related problems. This led to the low-degree conjecture, which predicts that when the low-degree advantage between a planted distribution and a uniform null distribution remains bounded, no efficient distinguisher can succeed after independent noise, provided that the planted distribution has permutation symmetry. Several works have produced counterexamples to variants of this conjecture or to versions for algorithms with higher time complexity, but the conjecture remained open in its standard binary, polynomial-time formulation. We disprove the polynomial-time low-degree conjecture by giving a family of examples in this setting. For every fixed integer $r\geq3$, we construct a permutation-invariant distribution $\mathbb{P}_n$ on simple graphs, with $\mathbb{Q}_n=G(n,1/2)$, such that every marginal of $\mathbb{P}_n$ on at most $D_n=\Theta((\log n)^{r-1})$ edges is uniform. Therefore, the low-degree advantage is zero through degree $D_n$. Nevertheless, after every edge is independently resampled at a fixed positive rate, a deterministic rank test strongly distinguishes the resulting distribution from $\mathbb{Q}_n$ in polynomial time. The construction chooses a subspace of a Reed--Muller code whose nonzero polynomials have small absolute bias, selects points whose evaluation vectors have no short linear dependencies, and evaluates a random alternating bilinear form on pairs of these vectors. Our result shows that low-degree indistinguishability, a uniform null distribution, permutation invariance, and independent resampling do not by themselves imply polynomial-time hardness, and suggests that a valid general conjecture must impose an additional condition.

cs.CC↗

arXiv2026

High-Rate Public-Key Pseudorandom Codes for Edit Errors

Pseudorandom codes (PRCs), introduced by Christ and Gunn (CRYPTO '2024), are error-correcting codes whose codewords are computationally indistinguishable from uniformly random strings, while still being decodable by someone holding the key. They provide a natural primitive for robust and undetectable watermarking, particularly in applications to AI-generated content. Although recent works have obtained strong results for substitution errors, the edit-error setting remains much less understood, especially in the high-rate regime and over small alphabets. We study public-key pseudorandom codes against edit errors. First, we give a new reduction showing that binary zero-bit PRCs robust against a constant fraction of substitution errors can be transformed into binary zero-bit PRCs robust against edit errors. Consequently, under any assumption that yields zero-bit Hamming-robust PRCs, one also obtains zero-bit PRCs for edit channels, albeit only for the weaker class of sublinear polynomial edit channels, namely channels with edit error rate $1/n^{\gamma}$ for any constant $\gamma>0$. In the high-rate regime, we construct public-key PRCs with rate arbitrarily close to $1$ over sufficiently large constant alphabets, and with rate arbitrarily close to $1/2$ over the binary alphabet. Moreover, if we allow the alphabet size to be $\mathrm{poly}(\lambda)$, where $\lambda$ is the security parameter, then our public-key PRCs can attain the Singleton bound for insertion-deletion channels. Taken together, these results yield the first high-rate public-key binary PRC constructions for edit channels, under the same assumption that yields zero-bit Hamming-robust PRCs.

cs.CR↗

arXiv2026

Near Optimal Algorithms for Noisy $k$-XOR under Low-Degree Heuristic

Noisy $k$-XOR is a basic average-case inference problem in which one observes random noisy $k$-ary parity constraints and seeks to recover, or more weakly, detect, a hidden Boolean assignment. A central question is to characterize the tradeoff among sample complexity, noise level, and running time. We give a recovery algorithm, and hence also a detection algorithm, for noisy $k$-XOR in the high-noise regime. For every parameter $D$, our algorithm runs in time $n^{D+O(1)}$ and succeeds whenever $$ m \ge C_k \frac{n^{k/2}}{D^{\,k/2-1}\delta^2}, $$ where $C_k$ is an explicit constant depending only on $k$, and $\delta$ is the noise bias. Our result matches the best previously known time--sample tradeoff for detection, while simultaneously yielding recovery guarantees. In addition, the dependence on the noise bias $\delta$ is optimal up to constant factors, matching the information-theoretic scaling. We also prove matching low-degree lower bounds. In particular, we show that the degree-$D$ low-degree likelihood ratio has bounded $L^2$-norm below the same threshold, up to the same factor $D^{k/2-1}$. Under the low-degree heuristic, this implies that our algorithm is near-optimal over a broad range of parameters. Our approach combines a refined second-moment analysis with color coding and dynamic programming for structured hypergraph embedding statistics. These techniques may be of independent interest for other average-case inference problems.

cs.CC↗

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↗

arXiv2024

Improved Explicit Near-Optimal Codes in the High-Noise Regimes

We study uniquely decodable codes and list decodable codes in the high-noise regime, specifically codes that are uniquely decodable from $\frac{1-\varepsilon}{2}$ fraction of errors and list decodable from $1-\varepsilon$ fraction of errors. We present several improved explicit constructions that achieve near-optimal rates, as well as efficient or even linear-time decoding algorithms. Our contributions are as follows. 1. Explicit Near-Optimal Linear Time Uniquely Decodable Codes: We construct a family of explicit $\mathbb{F}_2$-linear codes with rate $\Omega(\varepsilon)$ and alphabet size $2^{\mathrm{poly} \log(1/\varepsilon)}$, that are capable of correcting $e$ errors and $s$ erasures whenever $2e + s < (1 - \varepsilon)n$ in linear-time. 2. Explicit Near-Optimal List Decodable Codes: We construct a family of explicit list decodable codes with rate $\Omega(\varepsilon)$ and alphabet size $2^{\mathrm{poly} \log(1/\varepsilon)}$, that are capable of list decoding from $1-\varepsilon$ fraction of errors with a list size $L = \exp\exp\exp(\log^{\ast}n)$ in polynomial time. 3. List Decodable Code with Near-Optimal List Size: We construct a family of explicit list decodable codes with an optimal list size of $O(1/\varepsilon)$, albeit with a suboptimal rate of $O(\varepsilon^2)$, capable of list decoding from $1-\varepsilon$ fraction of errors in polynomial time. Furthermore, we introduce a new combinatorial object called multi-set disperser, and use it to give a family of list decodable codes with near-optimal rate $\frac{\varepsilon}{\log^2(1/\varepsilon)}$ and list size $\frac{\log^2(1/\varepsilon)}{\varepsilon}$, that can be constructed in probabilistic polynomial time and decoded in deterministic polynomial time. We also introduce new decoding algorithms that may prove valuable for other graph-based codes.

cs.IT↗

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↗

arXiv2021

Nilpotent Category of Abelian Category and Self-Adjoint Functors

Let $\mathcal{C}$ be an additive category. The nilpotent category $\mathrm{Nil} (\mathcal{C})$ of $\mathcal{C}$, consists of objects pairs $(X, x)$ with $X\in\mathcal{C}, x\in\mathrm{End}_{\mathcal{C}}(X)$ such that $x^n=0$ for some positive integer $n$, and a morphism $f:(X, x)\rightarrow (Y,y)$ is $f\in \mathrm{Hom}_{\mathcal{C}}(X, Y)$ satisfying $fx=yf$. A general theory of $\mathrm{Nil}(\mathcal{C})$ is established and it is abelian in the case that $\mathcal{C}$ is abelian. Two abelian categories are equivalent if and only if their nilpotent categories are equivalent, which generalizes a Song, Wu, and Zhang's result. As an application, it is proved all self-adjoint functors are naturally isomorphic to $\mathrm{Hom}$ and $\mathrm{Tensor}$ functors over the category $\mathrm{Nil}$ of finite-dimensional vector spaces. Both $\mathrm{Hom}$ and $\mathrm{Tensor}$ can be naturally generalized to $\mathrm{HOM}$ and $\mathrm{Tensor}$ functor over $\mathrm{Nil}(\mathcal{V})$. They are still self-adjoint, but intrinsically different.

math.CT↗

arXiv2021

A Generalized Determinant of Matrices and Applications

A generalized definition of the determinant of matrices is given, which is compatible with the usual determinant for square matrices and keeps many important properties, such as being an alternating multilinear function, keeping multiplication formula and partly keeping the Cauchy-Binet's formula. As applications of the new theory, the generalized Cramer's rule and the generalized oriented volume are obtained.

math.CA↗