construct equiangular tight frames

This paper systematically extends finite tight frame theory to the quaternionic vector space ℍᵈ, addressing the existence, construction, and classification of equiangular lines—i.e., equiangular and equidimensional subspaces. Methodologically, it integrates quaternionic linear algebra, representation theory of Lie groups, and projection operator theory to establish variational characterizations of quaternionic tight frames, develop group-based frame constructions, and derive projective/Unitary equivalence criteria. It formulates the first quaternionic analogue of Zauner’s conjecture and constructs a reversible migration and symmetric mapping mechanism among equiangular configurations in ℝᵈ, ℂᵈ, and ℍᵈ. Key contributions include: (i) establishing a complete theoretical foundation for quaternionic tight frames; (ii) identifying necessary dimensional constraints for the existence of equiangular lines in ℍᵈ; and (iii) providing a unified analytical and transformational framework for equiangular structures across the three number fields—thereby filling a fundamental theoretical gap in the field.

This paper investigates the binder structure of doubly transitive equiangular tight frames (DTETFs). The central problem is to determine whether the binder is empty or forms a balanced incomplete block design (BIBD). Employing tools from group actions, symplectic and quadratic forms over finite fields, combinatorial design theory, and Naimark complements, we establish—rigorously for the first time—that the binder of any DTETF must either be empty or constitute a BIBD. Furthermore, we show that the binder of its Naimark complement corresponds precisely to an oval in the associated BIBD, thereby forging novel connections between DTETFs, oval geometry, and affine Lagrangian subspaces. Finally, we completely characterize the binder properties of four families of binary quadratic-form-based DTETFs: two families admit nonempty binders only in finitely many exceptional cases, while the other two always yield BIBDs. This work unifies and generalizes existing ETF constructions based on symplectic forms and quadratic forms.

This work addresses the algebraic construction of equiangular tight frames (ETFs) over finite fields. To this end, it introduces the Galois inner product to establish a novel framework theory—defining Galois frames, Galois Gram matrices, and Galois frame operators. The core method establishes a necessary and sufficient condition under which Galois self-dual codes induce Galois ETFs, and explicitly constructs multiple families of Galois ETFs from Galois self-dual quasi-cyclic codes for the first time. These results unify and generalize classical Euclidean and Hermitian ETF constructions, deeply integrating algebraic coding theory—particularly self-duality—into finite-field frame design. The proposed framework provides new algebraic tools and systematic construction methods for structured measurement matrices in low-dimensional embeddings, compressed sensing, and quantum information theory.

This paper addresses dictionary design for block-sparse signal recovery in compressed sensing, focusing on equi-isoclinic tight fusion frames (EITFFs) whose subspace dimension equals half the ambient dimension, with the goal of minimizing block coherence to suppress information aliasing. Method: Leveraging projective geometry, group action analysis, and optimization theory for tight fusion frames, we establish necessary and sufficient existence conditions and explicitly construct infinitely many families of highly symmetric EITFFs. Contribution/Results: We provide the first complete characterization of the existence and algebraic structure of half-dimensional EITFFs, revealing their fundamental connection to the Radon–Hurwitz matrix theory and proving that all such configurations exhibit even-permutation full symmetry. The proposed dictionaries significantly enhance robustness against noise and undersampling, and establish both a theoretical foundation and a constructive paradigm for optimal Grassmannian coding.

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.

本文利用框架理论和压缩感知工具，建立了神经网络叠加的数学理论，并证明了在不同条件下特征恢复的可能性。

本文提出了一种新的表示论框架——蜂巢层次结构，用于提供R_2(δ)的渐近上界，通过保留每两个行不可约表示和每个坐标框转移通道，并结合移动投影定理，得到一个显式的四参数指数κ_HC。

This work investigates the generalized covering radius of linear codes from a geometric perspective, introducing for the first time the notion of $(\rho,t)$-saturating sets as their counterpart in finite geometry, thereby unifying classical saturating sets and $t$-strong blocking sets. By integrating tools from finite geometry, the dual Grassmannian criterion, affine methods, and combinatorial configuration techniques, the paper establishes novel connections between coding theory and finite geometry. The main contributions include multiple equivalent characterizations and lower bounds on the size of $(\rho,t)$-saturating sets, along with efficient construction methods derived from strong blocking sets, graphs, and projective configurations. These results provide a systematic framework for the analysis and application of generalized covering radii in coding theory.

This work investigates optimal quantum encoding strategies for classical data in the context of quantum-assisted statistical inference. It introduces maximal quantum leakage as a task-agnostic, universal metric for encoding quality and combines information-theoretic analysis, tight frame theory, and quantum measurement techniques to demonstrate that pure-state encodings achieve optimality via phase encoding under resource constraints and via basis encoding in resource-rich regimes. The study establishes, for the first time, the universality of maximal quantum leakage and reveals the unique symmetric optimality of equiangular tight frames (ETFs) in low-dimensional systems, linking them to fundamental structures such as symmetric informationally complete positive operator-valued measures (SIC-POVMs). Numerical experiments corroborate the theoretical findings and clarify the optimal encoding forms across varying resource conditions.

本文解决了Khatri-Rao结构矩阵在子空间嵌入问题中的性能分析，通过证明仅需m = Õ(k/ε^2)维度即可达到(1±ε)误差的嵌入效果，改善了之前的研究结果。