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Frédérique Oggier

Publications and source records attributed to Frédérique Oggier.

At least 19 recordsLinked to original sources
arXiv2026

Real Fiducial Vectors Only Exist in Dimensions $d\equiv 3 \pmod{4}$

We investigate restrictions on the dimensions and coordinate fields of real Weyl-Heisenberg fiducial vectors. Our main result is that such vectors can exist only in dimensions $d\equiv 3\pmod{4}$. This result is motivated by evidence showing that real Weyl-Heisenberg fiducial vectors can exist only in dimensions $d = 4m^2 + 3, m \in \mathbb{Z}_{\geq 0}$. We further restrict the possible dimensions when the coordinate field is a totally real number field, an even or odd degree extension of $\mathbb{Q}$, a real quadratic field, or a maximal real subfield of a cyclotomic field. Lastly, we establish non-existence over odd-degree number fields under either positivity of all non-trivial cyclic-shift overlaps or a conjectural containment of a distinguished quadratic field in the SIC field.

math.NT↗
arXiv2026

On the Existence of Algebraic Equiangular Lines

We consider real and complex equiangular lines, generated by unit vectors. We show that, for an arbitrary dimension $d$, if there exists a set of $d^2$ equiangular unit vectors in $\mathbb{C}^d$, then there must exist a set of $d^2$ equiangular unit vectors with all of their coefficients in a number field. This result is motivated by the question of constructing SIC-POVMs in quantum physics and conjectures around them. We discuss applications of our techniques to the case of real equiangular lines and consequences of the above results.

quant-ph↗
arXiv2025

About the Rankin and Bergé-Martinet Constants from a Coding Theory View Point

The Rankin constant $γ_{n,l}$ measures the largest volume of the densest sublattice of rank $l$ of a lattice $Λ\in \RR^n$ over all such lattices of rank $n$. The Bergé-Martinet constant $γ'_{n,l}$ is a variation that takes into account the dual lattice. Exact values and bounds for both constants are mostly open in general. We consider the case of lattices built from linear codes, and look at bounds on $γ_{n,l}$ and $γ'_{n,l}$. In particular, we revisit known results for $n=3,4,5,8$ and give lower and upper bounds for the open cases $γ_{5,2},γ_{7,2}$ and $γ'_{5,2},γ'_{7,2}$.

cs.IT↗
arXiv2020

Coding Constructions for Efficient Oblivious Transfer from Noisy Channels

We consider oblivious transfer protocols performed over binary symmetric channels in a malicious setting where parties will actively cheat if they can. We provide constructions purely based on coding theory that achieve an explicit positive rate, the essential ingredient being the existence of linear codes whose Schur products are asymptotically good.

cs.IT↗
arXiv2020

A Quadratic Form Approach to Construction A of Lattices over Cyclic Algebras

We propose a construction of lattices from (skew-) polynomial codes, by endowing quotients of some ideals in both number fields and cyclic algebras with a suitable trace form. We give criteria for unimodularity. This yields integral and unimodular lattices with a multiplicative structure. Examples are provided.

cs.IT↗
arXiv2017

Performance of Lattice Coset Codes on Universal Software Radio Peripherals

We consider an experimental setup of three USRPs that implement a wiretap channel, two USRPs are the legitimate players Alice and Bob, while the third USRP is the eavesdropper, whose position we vary to evaluate information leakage. The experimented channels are close to slow fading channels, and coset coding of lattice constellations is used for transmission, allowing to introduce controlled randomness at the transmitter. Simulation and measurement results show to which extent coset coding can provide confidentiality, as a function of Eve's position, and the amount of randomness used.

cs.IT↗
arXiv2016

Modular Lattices from a Variation of Construction A over Number Fields

We consider a variation of Construction A of lattices from linear codes based on two classes of number fields, totally real and CM Galois number fields. We propose a generic construction with explicit generator and Gram matrices, then focus on modular and unimodular lattices, obtained in the particular cases of totally real, respectively, imaginary, quadratic fields. Our motivation comes from coding theory, thus some relevant properties of modular lattices, such as minimal norm, theta series, kissing number and secrecy gain are analyzed. Interesting lattices are exhibited.

cs.IT↗
arXiv2015

On skew polynomial codes and lattices from quotients of cyclic division algebras

We propose a variation of Construction A of lattices from linear codes defined using the quotient $Λ/\mathfrak pΛ$ of some order $Λ$ inside a cyclic division $F$-algebra, for $\mathfrak p$ a prime ideal of a number field $F$. To obtain codes over this quotient, we first give an isomorphism between $Λ/\mathfrak pΛ$ and a ring of skew polynomials. We then discuss definitions and basic properties of skew polynomial codes, which are needed for Construction A, but also explore further properties of the dual of such codes. We conclude by providing an application to space-time coding, which is the original motivation to consider cyclic division $F$-algebras as a starting point for this variation of Construction A.

cs.IT↗
arXiv2015

Compressed Differential Erasure Codes for Efficient Archival of Versioned Data

In this paper, we study the problem of storing an archive of versioned data in a reliable and efficient manner in distributed storage systems. We propose a new storage technique called differential erasure coding (DEC) where the differences (deltas) between subsequent versions are stored rather than the whole objects, akin to a typical delta encoding technique. However, unlike delta encoding techniques, DEC opportunistically exploits the sparsity (i.e., when the differences between two successive versions have few non-zero entries) in the updates to store the deltas using compressed sensing techniques applied with erasure coding. We first show that DEC provides significant savings in the storage size for versioned data whenever the update patterns are characterized by in-place alterations. Subsequently, we propose a practical DEC framework so as to reap storage size benefits against not just in-place alterations but also real-world update patterns such as insertions and deletions that alter the overall data sizes. We conduct experiments with several synthetic workloads to demonstrate that the practical variant of DEC provides significant reductions in storage overhead (up to 60\% depending on the workload) compared to baseline storage system which incorporates concepts from Rsync, a delta encoding technique to store and synchronize data across a network.

cs.IT↗
arXiv2014

Construction A of Lattices over Number Fields and Block Fading Wiretap Coding

We propose a lattice construction from totally real and CM fields, which naturally generalizes the Construction A of lattices from $p$-ary codes obtained from the cyclotomic field $\mathbb{Q}(ζ_p)$, $p$ a prime, which in turn contains the so-called Construction A of lattices from binary codes as a particular case. We focus on the maximal totally real subfield $\mathbb{Q}(ζ_{p^r}+ζ_{p}^{-r})$ of the cyclotomic field $\mathbb{Q}(ζ_{p^r})$, $r\geq 1$. Our construction has applications to coset encoding of algebraic lattice codes, and we detail the case of coset encoding of block fading wiretap codes.

cs.IT↗
arXiv2014

Sparsity Exploiting Erasure Coding for Resilient Storage and Efficient I/O Access in Delta based Versioning Systems

In this paper we study the problem of storing reliably an archive of versioned data. Specifically, we focus on systems where the differences (deltas) between subsequent versions rather than the whole objects are stored - a typical model for storing versioned data. For reliability, we propose erasure encoding techniques that exploit the sparsity of information in the deltas while storing them reliably in a distributed back-end storage system, resulting in improved I/O read performance to retrieve the whole versioned archive. Along with the basic techniques, we propose a few optimization heuristics, and evaluate the techniques' efficacy analytically and with numerical simulations.

cs.IT↗
arXiv2014

Rank weight hierarchy of some classes of cyclic codes

We study the rank weight hierarchy, thus in particular the rank metric, of cyclic codes over the finite field $\mathbb F_{q^m}$, $q$ a prime power, $m \geq 2$. We establish the rank weight hierarchy for $[n,n-1]$ cyclic codes and characterize $[n,k]$ cyclic codes of rank metric 1 when (1) $k=1$, (2) $n$ and $q$ are coprime, and (3) the characteristic $char(\mathbb F_q)$ divides $n$. Finally, for $n$ and $q$ coprime, cyclic codes of minimal $r$-rank are characterized, and a refinement of the Singleton bound for the rank weight is derived.

cs.IT↗
arXiv2014

Connections Between Construction D and Related Constructions of Lattices

Most practical constructions of lattice codes with high coding gains are multilevel constructions where each level corresponds to an underlying code component. Construction D, Construction D$'$, and Forney's code formula are classical constructions that produce such lattices explicitly from a family of nested binary linear codes. In this paper, we investigate these three closely related constructions along with the recently developed Construction A$'$ of lattices from codes over the polynomial ring $\mathbb{F}_2[u]/u^a$. We show that Construction by Code Formula produces a lattice packing if and only if the nested codes being used are closed under Schur product, thus proving the similarity of Construction D and Construction by Code Formula when applied to Reed-Muller codes. In addition, we relate Construction by Code Formula to Construction A$'$ by finding a correspondence between nested binary codes and codes over $\mathbb{F}_2[u]/u^a$. This proves that any lattice constructible using Construction by Code Formula is also constructible using Construction A$'$. Finally, we show that Construction A$'$ produces a lattice if and only if the corresponding code over $\mathbb{F}_2[u]/u^a$ is closed under shifted Schur product.

cs.IT↗
arXiv2013

Locally Repairable Codes with Multiple Repair Alternatives

Distributed storage systems need to store data redundantly in order to provide some fault-tolerance and guarantee system reliability. Different coding techniques have been proposed to provide the required redundancy more efficiently than traditional replication schemes. However, compared to replication, coding techniques are less efficient for repairing lost redundancy, as they require retrieval of larger amounts of data from larger subsets of storage nodes. To mitigate these problems, several recent works have presented locally repairable codes designed to minimize the repair traffic and the number of nodes involved per repair. Unfortunately, existing methods often lead to codes where there is only one subset of nodes able to repair a piece of lost data, limiting the local repairability to the availability of the nodes in this subset. In this paper, we present a new family of locally repairable codes that allows different trade-offs between the number of contacted nodes per repair, and the number of different subsets of nodes that enable this repair. We show that slightly increasing the number of contacted nodes per repair allows to have repair alternatives, which in turn increases the probability of being able to perform efficient repairs. Finally, we present pg-BLRC, an explicit construction of locally repairable codes with multiple repair alternatives, constructed from partial geometries, in particular from Generalized Quadrangles. We show how these codes can achieve practical lengths and high rates, while requiring a small number of nodes per repair, and providing multiple repair alternatives.

cs.IT↗
arXiv2013

2- and 3-modular Lattice Wiretap Codes in Small Dimensions

A recent line of work on lattice codes for Gaussian wiretap channels introduced a new lattice invariant called secrecy gain as a code design criterion which captures the confusion that lattice coding produces at an eavesdropper. Following up the study of unimodular lattice wiretap codes [1], this paper investigates 2- and 3-modular lattices and compares them with unimodular lattices. Most even 2- and 3-modular lattices are found to have better performance, that is, a higher secrecy gain than the best unimodular lattices in dimension n, n is between 2 and 23. Odd 2-modular lattices are considered, too, and three lattices are found to outperform the best unimodular lattices.

math.NT↗
arXiv2013

In-Network Redundancy Generation for Opportunistic Speedup of Backup

Erasure coding is a storage-efficient alternative to replication for achieving reliable data backup in distributed storage systems. During the storage process, traditional erasure codes require a unique source node to create and upload all the redundant data to the different storage nodes. However, such a source node may have limited communication and computation capabilities, which constrain the storage process throughput. Moreover, the source node and the different storage nodes might not be able to send and receive data simultaneously -- e.g., nodes might be busy in a datacenter setting, or simply be offline in a peer-to-peer setting -- which can further threaten the efficacy of the overall storage process. In this paper we propose an "in-network" redundancy generation process which distributes the data insertion load among the source and storage nodes by allowing the storage nodes to generate new redundant data by exchanging partial information among themselves, improving the throughput of the storage process. The process is carried out asynchronously, utilizing spare bandwidth and computing resources from the storage nodes. The proposed approach leverages on the local repairability property of newly proposed erasure codes tailor made for the needs of distributed storage systems. We analytically show that the performance of this technique relies on an efficient usage of the spare node resources, and we derive a set of scheduling algorithms to maximize the same. We experimentally show, using availability traces from real peer-to-peer applications as well as Google data center availability and workload traces, that our algorithms can, depending on the environment characteristics, increase the throughput of the storage process significantly (up to 90% in data centers, and 60% in peer-to-peer settings) with respect to the classical naive data insertion approach.

cs.DC↗
arXiv2013

An Error Probability Approach to MIMO Wiretap Channels

We consider MIMO (Multiple Input Multiple Output) wiretap channels, where a legitimate transmitter Alice is communicating with a legitimate receiver Bob in the presence of an eavesdropper Eve, and communication is done via MIMO channels. We suppose that Alice's strategy is to use a codebook which has a lattice structure, which then allows her to perform coset encoding. We analyze Eve's probability of correctly decoding the message Alice meant to Bob, and from minimizing this probability, we derive a code design criterion for MIMO lattice wiretap codes. The case of block fading channels is treated similarly, and fast fading channels are derived as a particular case. The Alamouti code is carefully studied as an illustration of the analysis provided.

cs.IT↗