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The Learning Parity with Noise (LPN) problem underlies several classic cryptographic primitives. Researchers have endeavored to demonstrate the algorithmic difficulty of this problem by attempting to find a reduction from the decoding problem of linear codes, for which several hardness results exist. Earlier studies used code smoothing as a technical tool to achieve such reductions, showing that they are possible for codes with vanishing rate. This has left open the question of attaining a reduction with positive-rate codes. Addressing this case, we characterize the efficiency of the reduction in terms of the parameters of the decoding and LPN problems. As a conclusion, we isolate the parameter regimes for which a meaningful reduction is possible and the regimes for which its existence is unlikely.

We consider regenerating codes in distributed storage systems where connections between the nodes are constrained by a graph. In this problem, the failed node downloads the information stored at a subset of vertices of the graph for the purpose of recovering the lost data. Compared to the standard setting, regenerating codes on graphs address two additional features. The repair information is moved across the network, and the cost of node repair is determined by the graphical distance from the helper nodes to the failed node. Accordingly, the helpers far away from the failed node may be expected to contribute less data for repair than the nodes in the neighborhood of that node. We analyze regenerating codes with nonuniform download for repair on graphs. Moreover, in the process of repair, the information moved from the helpers to the failed node may be combined through intermediate processing, reducing the repair bandwidth. We derive lower bounds for communication complexity of node repair on graphs, including repair schemes with nonuniform download and intermediate processing, and construct codes that attain these bounds. Additionally, some of the nodes may act as adversaries, introducing errors into the data moved in the network. For repair on graphs in the presence of adversarial nodes, we construct codes that support node repair and error correction in systematic nodes., Comment: 29pp

We consider the problem of distilling uniform random bits from an unknown source with a given $p$-entropy using linear hashing. As our main result, we estimate the expected $p$-divergence from the uniform distribution over the ensemble of random linear codes for all integer $p\ge 2$. The proof relies on analyzing how additive noise, determined by a random element of the code from the ensemble, acts on the source distribution. This action leads to the transformation of the source distribution into an approximately uniform one, a process commonly referred to as distribution smoothing. We also show that hashing with Reed-Muller matrices reaches intrinsic randomness of memoryless Bernoulli sources in the $l_p$ sense for all integer $p\ge 2$.

We construct a new family of permutationally invariant codes that correct $t$ Pauli errors for any $t\ge 1$. We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases, which also admit transversal gates. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. Furthermore, our new code family includes a new $((4,2,2))$ optimal single-deletion-correcting code. As a separate result, we generalize the conditions for permutationally invariant codes to correct $t$ Pauli errors from the previously known results for $t=1$ to any number of errors. For small $t$, these conditions can be used to construct new examples of codes by computer., Comment: v3: Accepted to Quantum. This is the published version of the paper, formatted in the style of the journal

A storage code is an assignment of symbols to the vertices of a connected graph $G(V,E)$ with the property that the value of each vertex is a function of the values of its neighbors, or more generally, of a certain neighborhood of the vertex in $G$. In this work we introduce a new construction method of storage codes, enabling one to construct new codes from known ones via an interleaving procedure driven by resolvable designs. We also study storage codes on $\mathbb Z$ and ${\mathbb Z}^2$ (lines and grids), finding closed-form expressions for the capacity of several one and two-dimensional systems depending on their recovery set, using connections between storage codes, graphs, anticodes, and difference-avoiding sets.

The action of a noise operator on a code transforms it into a distribution on the respective space. Some common examples from information theory include Bernoulli noise acting on a code in the Hamming space and Gaussian noise acting on a lattice in the Euclidean space. We aim to characterize the cases when the output distribution is close to the uniform distribution on the space, as measured by R\'enyi divergence of order $\alpha \in (1,\infty]$. A version of this question is known as the channel resolvability problem in information theory, and it has implications for security guarantees in wiretap channels, error correction, discrepancy, worst-to-average case complexity reductions, and many other problems. Our work quantifies the requirements for asymptotic uniformity (perfect smoothing) and identifies explicit code families that achieve it under the action of the Bernoulli and ball noise operators on the code. We derive expressions for the minimum rate of codes required to attain asymptotically perfect smoothing. In proving our results, we leverage recent results from harmonic analysis of functions on the Hamming space. Another result pertains to the use of code families in Wyner's transmission scheme on the binary wiretap channel. We identify explicit families that guarantee strong secrecy when applied in this scheme, showing that nested Reed-Muller codes can transmit messages reliably and securely over a binary symmetric wiretap channel with a positive rate. Finally, we establish a connection between smoothing and error correction in the binary symmetric channel., Comment: 30 pages, 2 figures. In v.2, the characterization of smoothing capacity was extended from integer orders of the R\'enyi divergence to all real orders $\alpha\in (1,\infty]$. Presentation reorganized, minor errors are corrected

We introduce a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes. We apply this framework to bosonic coding, obtaining multimode extensions of the cat codes that can outperform previous constructions while requiring a similar type of overhead. Our polytope-based cat codes consist of sets of points with large separation that at the same time form averaging sets known as spherical designs. We also recast concatenations of CSS codes with cat codes as quantum spherical codes, revealing a new way to autonomously protect against dephasing noise., Comment: 5 + 12 pages, 3 figures, 5 tables

A storage code on a graph $G$ is a set of assignments of symbols to the vertices such that every vertex can recover its value by looking at its neighbors. We consider the question of constructing large-size storage codes on triangle-free graphs constructed as coset graphs of binary linear codes. Previously it was shown that there are infinite families of binary storage codes on coset graphs with rate converging to 3/4. Here we show that codes on such graphs can attain rate asymptotically approaching 1. Equivalently, this question can be phrased as a version of hat-guessing games on graphs (e.g., P.J. Cameron e.a., \emph{Electronic J. Comb.} 2016). In this language, we construct triangle-free graphs with success probability of the players approaching one as the number of vertices tends to infinity. Furthermore, finding linear index codes of rate approaching zero is also an equivalent problem. Another family of storage codes on triangle-free graphs of rate approaching 1 was constructed earlier by A. Golovnev and I. Haviv (36th Computational Complexity Conf., 2021) relying on a different family of graphs., Comment: Final version, to appear in Combinatorics

We continue our study of regenerating codes in distributed storage systems where connections between the nodes are constrained by a graph. In this problem, the failed node downloads the information stored at a subset of vertices of the graph for the purpose of recovering the lost data. This information is moved across the network, and the cost of node repair is determined by the graphical distance from the helper nodes to the failed node. This problem was formulated in our recent work (IEEE IT Transactions, May 2022) where we showed that processing of the information at the intermediate nodes can yield savings in repair bandwidth over the direct forwarding of the data. While the previous paper was limited to the MSR case, here we extend our study to the case of general regenerating codes. We derive a lower bound on the repair bandwidth and formulate repair procedures with intermediate processing for several families of regenerating codes, with an emphasis on the recent constructions from multilinear algebra. We also consider the task of data retrieval for codes on graphs, deriving a lower bound on the communication bandwidth and showing that it can be attained at the MBR point of the storage-bandwidth tradeoff curve.

Soft and flexible conductors are essential in the development of soft robots, wearable electronics, as well as electronic tissue and implants. However, conventional soft conductors are inherently characterized by a large change in conductance upon mechanical deformation or under alternating environmental conditions, e.g., humidity, drastically limiting their application potential and performance. Here, we demonstrate a novel concept for the development of strain-invariant, fatigue resistant and highly water stable soft conductor. By combining different thin film technologies in a three-dimensional fashion, we develop nano- and micro-engineered, multi-layered (< 50 nm), ultra-lightweight (< 15 mg/cm$^3$) foam-like composite framework structures based on PEDOT:PSS and PTFE. The all-organic composite framework structures are characterized by conductivities of up to 184 S/m, remaining strain-invariant between 80 % compressive and 25 % tensile strain. We further show, that the multi-layered composites are characterized by properties that surpass that of framework structures based on the individual materials. Both, the initial electrical and mechanical properties of the composite framework structures are retained during long-term cycling, even after 2000 cycles at 50 % compression. Furthermore, the PTFE functionalization renders the framework structure highly hydrophobic, resulting in stable electrical properties, even when immersed in water for up to 30 days. The here presented concept overcomes the previous limitations of strain-invariant soft conductors and demonstrates for the first time a versatile approach for the development of innovative multi-scaled and multi-layered functional materials, for applications in soft electronics, energy storage and conversion, sensing, catalysis, water and air purification, as well as biomedicine., Comment: Corresponding authors: Dr.-Ing. Fabian Sch\"utt (fas@tf.uni-kiel.de) and Dr.-Ing. Stefan Schr\"oder (ssch@tf.uni-kiel.de); Shared senior authorship: Dr.-Ing. Fabian Sch\"utt and Dr.-Ing. Stefan Schr\"oder

We address the maximum size of binary codes and binary constant weight codes with few distances. Previous works established a number of bounds for these quantities as well as the exact values for a range of small code lengths. As our main results, we determine the exact size of maximal binary codes with two distances for all lengths $n\ge 6$ as well as the exact size of maximal binary constant weight codes with 2,3, and 4 distances for several values of the weight and for all but small lengths., Comment: Main text 23 pp. and Appendix 17pp

We study the maximum cardinality problem of a set of few distances in the Hamming and Johnson spaces. We formulate semidefinite programs for this problem and extend the 2011 works by Barg-Musin and Musin-Nozaki. As our main result, we find new parameters for which the maximum size of two- and three-distance sets is known exactly., Comment: Renew the whole article to add more results and authors

As renewable energy is becoming a critical energy source to meet the global demand, electrochemical energy storage devices become indispensable for the efficient energy storage and reliable supply. The electrode material is the key factor determining the energy storage capacity and the power delivery of the devices. Carbon-based materials are emerging as a viable candidate for electrodes, yet their low energy densities impede the development of advanced energy storage materials. Nanoparticle decoration of the carbon structures is one of the most promising and easy-to-implement a strategy to enhance the charge-storage performance of carbon-based electrodes. Decoration by nanoparticles of metals, metal oxides, nitrides, carbides, phosphides, chalcogenides, and bimetallic components lead to significant enhancements in the structural and electronic properties, pore refinement, charge storage, and charge transfer kinetics of both pristine and doped carbon structures, thereby making their performance promising for next-generation energy storage devices. This review covers the state-of-art nanoparticle decorated nanocarbons for battery, supercapacitor, and metal-ion capacitor applications. A critical analysis of the elemental composition, structure, associated physico-chemical properties and performance relationships of nanoparticle-decorated nanocarbon electrodes is provided as well to inform the future development of the next generation of advanced energy storage materials, devices, and systems., Comment: 67 pages, 17 figures, 5 tables

Consider an assignment of bits to the vertices of a connected graph $G(V,E)$ with the property that the value of each vertex is a function of the values of its neighbors. A collection of such assignments is called a {\em storage code} of length $|V|$ on $G$. The storage code problem can be equivalently formulated as maximizing the probability of success in a {\em guessing game} on graphs, or constructing {\em index codes} of small rate. If $G$ contains many cliques, it is easy to construct codes of rate close to 1, so a natural problem is to construct high-rate codes on triangle-free graphs, where constructing codes of rate $>1/2$ is a nontrivial task, with few known results. In this work we construct infinite families of linear storage codes with high rate relying on coset graphs of binary linear codes. We also derive necessary conditions for such codes to have high rate, and even rate potentially close to one. We also address correction of multiple erasures in the codeword, deriving recovery guarantees based on expansion properties of the graph. Finally, we point out connections between linear storage codes and quantum CSS codes, a link to bootstrap percolation and contagion spread in graphs, and formulate a number of open problems., Comment: 20 pages. V2: minor corrections and additions