incidence matrix construction

This work addresses the long-standing challenge of explicitly constructing near-optimal lossless rank extractors, weak subspace designs, and strong $s$-blocking sets over small finite fields whose size depends only on the rank or codimension. By integrating tools from function field theory, polynomial identity testing, and Fourier analysis based on $\varepsilon$-biased sets, the authors achieve the first explicit near-optimal constructions of these objects over non-prime fields with $q \geq \mathrm{poly}(s)$. Notably, the resulting strong $s$-blocking set has size $O(s(k - s)q^s)$, improving upon the previous exponential bound $2^{O(s^2 \log s)} q^s k$ and matching the non-explicit optimal asymptotics. The paper also presents the first explicit near-optimal constructions for both lossless rank extractors and weak subspace designs in this setting.

This work resolves the central conjecture of Nelson and Nguyen (FOCS 2013) on the optimal dimension and sparsity of oblivious subspace embeddings (OSEs): For any $n ge d$ and $varepsilon ge d^{-O(1)}$, does there exist a random matrix $Pi$ such that, with high probability, $(1-varepsilon)|Ax| le |Pi Ax| le (1+varepsilon)|Ax|$ holds simultaneously for all $A in mathbb{R}^{n imes d}$ and $x in mathbb{R}^d$? We introduce an iterative decoupling technique for matrix concentration, overcoming the universality bottleneck of conventional higher-order moment analysis and enabling fine-grained control over trace moment bounds. Our approach yields the first OSE with embedding dimension $ ilde{O}(d/varepsilon^2)$—achieving both subpolynomial logarithmic factors and fast-matrix-multiplication structure. This improves upon prior constructions and significantly accelerates downstream applications such as large-scale linear regression.

This work investigates the Shortest Vector Problem (SVP) on ideal lattices over number fields, aiming to establish probabilistic bounds and asymptotic counting theorems for SVP over discrete families of ideal lattices. Methodologically, it unifies algebraic code lifting constructions with SVP analysis on ideal lattices for the first time, integrating algebraic number theory, lattice theory, probabilistic methods, and Rogers-type integral geometry techniques. It derives an asymptotic counting formula for algebraic integer matrices of fixed rank under Euclidean norm constraints and establishes a Rogers-type integral identity applicable to discrete families of ideal lattices. Key contributions include: (i) a tight probabilistic upper bound on the SVP length; (ii) a proof that a broad class of discrete ideal lattices inherits classical moment estimates and achieves optimal SVP bounds; and (iii) validation of the framework’s universality under algebraic lifting constructions—thereby providing foundational theoretical support for number-field-based lattice cryptography and algorithm design.

Characterizing threshold rates for local properties—such as list-decodability and list-recoverability—in coding theory, and extending these characterizations to subspace design codes. Method: We establish a rigorous equivalence between subspace design codes and random linear codes with respect to all local properties, proving that the former can fully inherit the local behavior of the latter with arbitrarily small rate loss. Leveraging this equivalence, we construct the first explicit folded Reed–Solomon codes and univariate multiplicity codes that simultaneously achieve all key local properties. We further extend the framework to matroid theory to obtain a deterministic polynomial-time algorithm for identifying correctable erasure patterns in maximally recoverable tensor codes. Contribution/Results: This work bridges the long-standing gap between random and explicit constructions for locally structured codes and introduces a unified algebraic-combinatorial paradigm for analyzing local properties, enabling precise threshold-rate characterizations and efficient deterministic verification of local correctness.

This work investigates the computational complexity of sparse vector search in real subspaces, lattices, and codes, establishing— for the first time over the reals—strong inapproximability results. Methodologically, it bypasses the PCP theorem and introduces a novel tensorized code product test, tightly integrating Littlewood–Offord theory with the Rudelson–Vershynin anti-concentration inequality, alongside kernel tensorization of ±1 random matrices and real embedding of quadratic equations. The main contribution is a proof that approximating the sparsest nonzero vector in a real subspace within any constant factor is NP-hard under randomized reductions. As a corollary, it yields the current best inapproximability bound for the Shortest Vector Problem (SVP) on lattices. This framework provides a unified hardness analysis paradigm for fundamental problems including real-domain sparse optimization, small-set expansion, and quantum separability.

This work addresses the explicit construction of variety-evasive subspace families—collections of subspaces that avoid all algebraic varieties from a given family—a problem of fundamental importance in pseudorandomness, hitting set constructions, and lossless rank condensers. By refining hitting set constructions based on Chow forms, we explicitly build subspace families in $n$-dimensional affine or projective space that evade all degree-$d$ algebraic varieties, with size nearly matching the theoretical lower bound. When $d = n^{1+\Omega(1)}$, our construction achieves a size exceeding the lower bound by only a polynomial factor, substantially improving upon prior work by Guo and yielding the best-known explicit construction to date.

This work proposes a unified framework for simultaneously addressing combinatorial and continuous sparsification problems—such as constraint satisfaction and spectral graph sparsification—centered on the sparse representational capacity of real-valued codes. By introducing Continuous-Valued Non-Redundancy Degree (CVNRD) as a key structural parameter, the study reveals that CVNRD serves both as an upper bound on achievable sparsification and as a fundamental lower-bound barrier for unbiased randomized sparsification methods. Leveraging an abstraction of real-valued codes, CVNRD-based analysis, and identification of approximate block-diagonal structures, the proposed approach achieves sparsification complexity nearly linear in CVNRD. Furthermore, the work establishes that CVNRD imposes a universal limitation across a broad class of randomized sparsification strategies.

This study addresses the problem of determining the shortest embeddings of linear codes with arbitrary hull dimensions under both Euclidean and Hermitian inner products. By leveraging the theory of quadratic forms over finite fields, classical group theory, and the congruence classification of Gram matrices, the authors develop a unified framework that generalizes prior results on shortest LCD codes and self-orthogonal embeddings to arbitrary hull dimensions and arbitrary finite fields. Within this framework, they completely characterize the minimal length of self-orthogonal embeddings for any given linear code, thereby improving upon existing bounds in the literature. Furthermore, through an explicit constructive algorithm, they obtain multiple new optimal linear codes that surpass the best-known entries in the BKLC database.

This work investigates the structure and metric properties of rank-metric codes from a geometric perspective. By introducing generator tensors and their slice spaces, the authors develop a unified geometric framework that translates the metric characteristics of codes into geometric conditions concerning hyperplane intersections, establishing a correspondence with equivalence classes of associated systems. They further propose a generalized notion of weights based on evasive systems, revealing novel connections between rank-metric codes and additive Hamming-metric codes, and enabling a unified derivation of several classical bounds and semifield-theoretic results. This framework effectively links rank distributions to the underlying geometric properties of systems, not only recovering known results—such as those for MRD and constant-rank codes—but also extending them, thereby demonstrating its efficacy in preserving and characterizing the intrinsic metric structure.

This work addresses the computational bottleneck in solving 4-block integer linear programs with large coefficients by introducing a novel approach based on an affine decomposition of the right-hand side. The method extends the existing n-fold integer programming algorithm to settings involving global variables. By leveraging key technical ingredients—including a vector rearrangement lemma, face guessing in hyperplane arrangements, dynamic high-multiplicity encoding, and faithful decomposition—the algorithm achieves, for the first time, polynomial dependence on the coefficient matrices A, B, and C. Consequently, the running time is significantly improved from the previous bound of \(n^{k^2 + O(1)}\) to \(f(k, \bar{\Delta}) \cdot n^{k + O(1)}\), where \(f\) depends only on the number of blocks \(k\) and the maximum subdeterminant \(\bar{\Delta}\), substantially outperforming the best-known prior results.