Lossless Dimension Expanders via Linearized Polynomials and Subspace Designs

For a vector space F^n over a field F, an (eta,beta)-dimension expander of degree d is a collection of d linear maps Gamma_j : F^n -> F^n such that for every subspace U of F^n of dimension at most eta n, the image of U under all the maps, sum_{j=1}^d Gamma_j(U), has dimension at least beta dim(U). Over a finite field, a random collection of d = O(1) maps Gamma_j offers excellent "lossless" expansion whp: beta ~~ d for eta >= Omega(1/d). When it comes to a family of explicit constructions (for growing n), however, achieving even modest expansion factor beta = 1+epsilon with constant degree is a non-trivial goal. We present an explicit construction of dimension expanders over finite fields based on linearized polynomials and subspace designs, drawing inspiration from recent progress on list-decoding in the rank-metric. Our approach yields the following: - Lossless expansion over large fields; more precisely beta >= (1-epsilon)d and eta >= (1-epsilon)/d with d = O_epsilon(1), when |F| >= Omega(n). - Optimal up to constant factors expansion over fields of arbitrarily small polynomial size; more precisely beta >= Omega(delta d) and eta >= Omega(1/(delta d)) with d=O_delta(1), when |F| >= n^{delta}. Previously, an approach reducing to monotone expanders (a form of vertex expansion that is highly non-trivial to establish) gave (Omega(1),1+Omega(1))-dimension expanders of constant degree over all fields. An approach based on "rank condensing via subspace designs" led to dimension expanders with beta >rsim sqrt{d} over large fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree

On subspace designs

Guruswami and Xing introduced subspace designs in 2013 to give the first construction of positive rate rank metric codes list-decodable beyond half the distance. In this paper we provide bounds involving the parameters of a subspace design, showing they are tight via explicit constructions. We point out a connection with sum-rank metric codes, dealing with optimal codes and minimal codes with respect to this metric. Applications to two-intersection sets with respect to hyperplanes, two-weight codes, cutting blocking sets and lossless dimension expanders are also provided

Applications of Derandomization Theory in Coding

Randomized techniques play a fundamental role in theoretical computer science and discrete mathematics, in particular for the design of efficient algorithms and construction of combinatorial objects. The basic goal in derandomization theory is to eliminate or reduce the need for randomness in such randomized constructions. In this thesis, we explore some applications of the fundamental notions in derandomization theory to problems outside the core of theoretical computer science, and in particular, certain problems related to coding theory. First, we consider the wiretap channel problem which involves a communication system in which an intruder can eavesdrop a limited portion of the transmissions, and construct efficient and information-theoretically optimal communication protocols for this model. Then we consider the combinatorial group testing problem. In this classical problem, one aims to determine a set of defective items within a large population by asking a number of queries, where each query reveals whether a defective item is present within a specified group of items. We use randomness condensers to explicitly construct optimal, or nearly optimal, group testing schemes for a setting where the query outcomes can be highly unreliable, as well as the threshold model where a query returns positive if the number of defectives pass a certain threshold. Finally, we design ensembles of error-correcting codes that achieve the information-theoretic capacity of a large class of communication channels, and then use the obtained ensembles for construction of explicit capacity achieving codes. [This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

Multipartite Quantum States and their Marginals

Subsystems of composite quantum systems are described by reduced density matrices, or quantum marginals. Important physical properties often do not depend on the whole wave function but rather only on the marginals. Not every collection of reduced density matrices can arise as the marginals of a quantum state. Instead, there are profound compatibility conditions -- such as Pauli's exclusion principle or the monogamy of quantum entanglement -- which fundamentally influence the physics of many-body quantum systems and the structure of quantum information. The aim of this thesis is a systematic and rigorous study of the general relation between multipartite quantum states, i.e., states of quantum systems that are composed of several subsystems, and their marginals. In the first part, we focus on the one-body marginals of multipartite quantum states; in the second part, we study general quantum marginals from the perspective of entropy.Comment: PhD thesis, ETH Zurich. The first part contains material from arXiv:1208.0365, arXiv:1204.0741, and arXiv:1204.4379. The second part is based on arXiv:1302.6990 and arXiv:1210.046

Lossless dimension expanders via linearized polynomials and subspace designs

For a vector space Fn over a field F, an (η, β)-dimension expander of degree d is a collection of d linear maps Γj: Fn→ Fn such that for every subspace U of Fn of dimension at most ηn, the image of U under all the maps, ∑j=1dΓj(U), has dimension at least α dim(U). Over a finite field, a random collection of d = O(1) maps Γj offers excellent “lossless” expansion whp: β≈d for η ≥ Ω(1/d). When it comes to a family of explicit constructions (for growing n), however, achieving even modest expansion factor β = 1+ ε with constant degree is a non-trivial goal. We present an explicit construction of dimension expanders over finite fields based on linearized polynomials and subspace designs, drawing inspiration from recent progress on list decoding in the rank metric. Our approach yields the following:Lossless expansion over large fields; more precisely β ≥ (1 − ε)d and η≥1−εd with d = Oε(1), when | F| ≥ Ω(n).Optimal up to constant factors expansion over fields of arbitrarily small polynomial size; more precisely β ≥ Ω(δd) and η ≥ Ω(1/(δd)) with d = Oδ(1), when | F| ≥ nδ. Previously, an approach reducing to monotone expanders (a form of vertex expansion that is highly non-trivial to establish) gave (Ω(1), 1 + Ω(1))-dimension expanders of constant degree over all fields. An approach based on “rank condensing via subspace designs” led to dimension expanders with β≳d over large finite fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree