In this paper, we consider the recently introduced concept of \emph{function-correcting codes (FCCs) with data protection}, which provide a certain level of error protection for the data and a higher level of protection for a desired function on the data. These codes are denoted by $(f\!:\!d_d,d_f)$-FCC, where $d_d$ is the minimum distance of the code and $d_f$ denotes the minimum distance between those codewords that correspond to different function values of a function $f:\mathbb{F}_q^k \to \mathrm{Im}(f)$, with $d_f \geq d_d$. We use a distance graph on a code based on the pairwise distances of its codewords, and show conditions under which a code cannot work as a \emph{strict} $(f\!:\!d_d,d_f)$-FCC, that is, code for which $d_f > d_d$. We then consider some well-known classes of codes, such as perfect codes and maximum distance separable (MDS) codes, and show that they cannot be used as \emph{strict} $(f\!:\!d_d,d_f)$-FCCs.

Function-correcting codes (FCCs) are designed to provide error protection for the value of a function computed on the data. Existing work typically focuses solely on protecting the function value and not the underlying data. In this work, we propose a general framework that offers protection for both the data and the function values. Since protecting the data inherently contributes to protecting the function value, we focus on scenarios where the function value requires stronger protection than the data itself. We first introduce a more general approach and a framework for function-correcting codes that incorporates data protection along with protection of function values. A two-step construction procedure for such codes is proposed, and bounds on the optimal redundancy of general FCCs with data protection are reported. Using these results, we exhibit examples that show that data protection can be added to existing FCCs without increasing redundancy. Using our two-step construction procedure, we present explicit constructions of FCCs with data protection for specific families of functions, such as locally bounded functions and the Hamming weight function. We associate a graph called minimum-distance graph to a code and use it to show that perfect codes and maximum distance separable (MDS) codes cannot provide additional protection to function values over and above the amount of protection for data for any function. Then we focus on linear FCCs and provide some results for linear functions, leveraging their inherent structural properties. To the best of our knowledge, this is the first instance of FCCs with a linear structure. Finally, we generalize the Plotkin and Hamming bounds well known in classical error-correcting coding theory to FCCs with data protection.

We introduce function-correcting partition codes (FCPCs), which are a natural generalization of function-correcting codes (FCCs). An FCPC is defined directly on a partition of the message space, rather than on a specific target function. We show that any FCC for a function $f$ is exactly an FCPC with respect to the domain partition induced by $f$, which makes these codes a natural generalization of FCCs. We use the join of domain partitions to construct a single code that protects multiple functions simultaneously. We define the notions of partition gains to measure the bandwidth saved by using a single FCPC for multiple functions instead of constructing separate FCCs for each function. We derive general lower and upper bounds on the redundancy of such FCPCs and illustrate the achievable gains through examples. We specialize this concept of using single code for protecting multiple functions to linear functions via coset partition of the intersection of their kernels. We also present explicit FCPC constructions for locally bounded partitions and grouped weight partitions. Then, we associate a partition graph with any given partition of $\mathbb{F}_q^k$, and show that the existence of a suitable clique in this graph yields a set of representative information vectors that achieves the optimal redundancy. Using the existence of a full-size clique in the weight partition and support partition, we obtain lower and upper bounds on the optimal redundancy of FCPCs for these partitions. We introduce the notion of a block-preserving contraction for a partition, which helps reduce the problem size of finding optimal redundancy for an FCPC. We further show that such a contraction exists for all weight-based partitions. Finally, we observe that FCPCs naturally provide a form of partial privacy in the sense that only the domain partition of the function needs to be revealed to the transmitter.

Function-correcting codes (FCCs) are designed to provide error protection for the value of a function computed on the data. Existing work typically focuses solely on protecting the function value and not the underlying data. In this work, we propose a general framework that offers protection for both the data and the function values. Since protecting the data inherently contributes to protecting the function value, we focus on scenarios where the function value requires stronger protection than the data itself. A two-step construction procedure for such codes is proposed, and bounds on the optimal redundancy of general FCCs with data protection are reported. Using these results, we exhibit examples that show that data protection can be added to existing FCCs without increasing redundancy. Using our two-step construction procedure, we present explicit constructions of FCCs with data protection for specific families of functions, such as locally bounded functions and the Hamming weight function. We associate a graph calledminimum-distance graphto a code and use it to show that perfect codes and maximum distance separable (MDS) codes cannot provide additional protection to function values over and above the amount of protection for data for any function. Then we focus on linear FCCs and provide some results for linear functions, leveraging their inherent structural properties. While FCCs for linear functions have been considered earlier in the literature, to the best of our knowledge, the linearity of the FCC itself has not been studied before. Finally, we generalize the Plotkin and Hamming bounds well known in classical error-correcting coding theory to FCCs with data protection.

The connection between secret sharing and matroid theory is well established. In this paper, we generalize the concepts of secret sharing and matroid ports to q-polymatroids. Specifically, we introduce the notion of an access structure on a vector space, and consider properties related to duality, minors, and the relationship to q-polymatroids. Finally, we show how rank-metric codes give rise to secret sharing schemes within this framework.

In federated learning, multiple parties train models locally and share their parameters with a central server, which aggregates them to update a global model. To address the risk of exposing sensitive data through local models, secure aggregation via secure multiparty computation has been proposed to enhance privacy. At the same time, perfect privacy can only be achieved by a uniform distribution of the "masked" local models to be aggregated. This raises a problem when working with real-valued data, as there is no measure on the reals that is invariant under the masking operation, and hence information leakage is bound to occur. Shifting the data to a finite field circumvents this problem, but as a downside runs into an inherent accuracy–complexity tradeoff issue due to fixed-point modular arithmetic as opposed to floating-point numbers that can simultaneously handle numbers of varying magnitudes. In this paper, a novel secure parameter aggregation method is proposed that employs the torus rather than a finite field. This approach guarantees perfect privacy for each party’s data by utilizing the uniform distribution on the torus, while avoiding accuracy losses. Experimental results show that the new protocol performs similarly to the model without secure aggregation while maintaining perfect privacy. Compared to the finite field secure aggregation, the torus-based protocol can in some cases significantly outperform it in terms of model accuracy and cosine similarity, hence making it a safer choice.

In this paper, we introduce a class of functions that assume only a limited number λ of values within a given Hamming ρ-ball and call them locally (ρ,λ)-bounded functions. We develop function-correcting codes (FCCs) for a subclass of these functions and propose an upper bound on the redundancy of FCCs. The bound is based on the minimum length of an error-correcting code with a given number of codewords and a minimum distance. Furthermore, we provide a sufficient optimality condition for FCCs when λ = 4. We also demonstrate that any function can be represented as a locally (ρ,λ)-bounded function, illustrating this with a representation of Hamming weight distribution functions. Furthermore, we present another construction of function-correcting codes for Hamming weight distribution functions.

This paper is eligible for the Jack Keil Wolf ISIT Student Paper Award. In this work, a flexible and robust private information retrieval (PIR) scheme based on binary non-maximum distance separable (non-MDS) codes is considered. This combines previous works on PIR schemes based on transitive non-MDS codes on one hand, and PIR from MDS-coded Byzantine and nonresponsive servers on the other hand. More specifically, a PIR scheme employing binary Reed–Muller (RM) codes tolerant to colluding, Byzantine, and non-responsive servers is constructed, and bounds for the achievable rates are derived under certain conditions. The construction of such schemes turns out to be much more involved than for MDS codes. Namely, the binary query vectors have to be selected with great care to hit the desired information sets, which is technically challenging as will be shown.

In this work, two practical concepts related to private information retrieval (PIR) are introduced and coinedfull support-rankPIR andstrongly linearPIR. Being of full support-rank is a technical, yet natural condition required to prove a converse result for a capacity expression and satisfied by almost all currently known capacity-achieving schemes, while strong linearity is a practical requirement enabling implementation over small finite fields with low subpacketization degree. Then, the capacity of MDS-coded, linear, full support-rank PIR in the presence of colluding servers is derived, as well as the capacity of symmetric, linear PIR with colluding, adversarial, and nonresponsive servers for the recently introduced concept of matched randomness. This positively settles the capacity conjectures stated by Freij-Hollantiet al.and Tajeddineet al.in the presented cases. It is also shown that, further restricting to strongly-linear PIR schemes with deterministic linear interference cancellation, the so-called star product scheme proposed by Freij-Hollantiet al.is essentially optimal and induces no capacity loss.

A private information retrieval (PIR) scheme allows a user to retrieve a file from a database without revealing any information on the file being requested. As of now, PIR schemes have been proposed for several kinds of storage systems, including replicated and MDS-coded systems. However, the problem of constructing PIR schemes on regenerating codes has been sparsely considered. A regenerating code is a storage code whose codewords are distributed among nodes, enabling efficient storage of files, as well as low-bandwidth retrieval of files and repair of nodes. Minimum-bandwidth regenerating (MBR) codes define a family of regenerating codes allowing a node repair with optimal bandwidth. Rashmi, Shah, and Kumar obtained a large family of MBR codes using the product-matrix (PM) construction. In this work, a new PIR scheme over PM-MBR codes is designed. The inherent redundancy of the PM structure is used to reduce the download communication complexity of the scheme. A lower bound on the PIR capacity of MBR-coded PIR schemes is derived, showing an interesting storage space vs. PIR rate trade-off compared to existing PIR schemes with the same reconstruction capability. The present scheme also outperforms a recent PM-MBR PIR construction of Dorkson and Ng.

We define and analyze low-rank parity-check (LRPC) codes over extension rings of the finite chain ring Zpr, where p is a prime and r is a positive integer. LRPC codes have originally been proposed by Gaborit et al. (2013) over finite fields for cryptographic applications. The adaption to finite rings is inspired by a recent paper by Kamche et al. (2019), which constructed Gabidulin codes over finite principle ideal rings with applications to space-time codes and network coding. We give a decoding algorithm based on simple linear-algebraic operations. Further, we derive an upper bound on the failure probability of the decoder. The upper bound is valid for errors whose rank is equal to the free rank.

In this work, we first prove the capacity of coded, linear symmetric private information retrieval (SPIR) in the presence of colluding, adversarial, and nonresponsive servers, giving a positive closure to the conjecture stated by Tajeddine et al. It is also shown that, further restricting to strongly-linear PIR schemes with linear interference cancellation, the so-called star product scheme proposed by Freij-Hollanti et al. is optimal. This observation enables to prove the capacity of strongly-linear (non-symmetric) PIR schemes for any number of files. Further, it also provides a positive proof in this practical special case for the conjectures stated in the asymptotic regime by Freij-Hollanti et al. and Tajeddine et al.

Private information retrieval (PIR) schemes for coded storage with colluding servers are presented, which are not restricted to maximum distance separable (MDS) codes. PIR schemes for general linear codes are constructed, and the resulting PIR rate is calculated explicitly. It is shown that codes with transitive automorphism groups yield the highest possible rates obtainable with the proposed scheme. In the special case of no server collusion, this rate coincides with the known asymptotic PIR capacity for MDS-coded storage systems. While many PIR schemes in the literature require field sizes that grow with the number of servers and files in the system, we focus especially on the case of a binary base field, for which Reed-Muller codes serve as an important and explicit class of examples.

Locally repairable codes (LRCs) have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others. Several works have thus been carried out to understand the optimal rate-distance tradeoff, but only recently the size of the alphabet has been taken into account. In this paper, a novel definition of locality is proposed to keep track of the precise number of nodes required for a local repair when the repair sets do not yield MDS codes. Then, a new alphabet-dependent bound is derived, which applies both to the new definition and the initial definition of locality. The new bound is based on consecutive residual codes and intrinsically uses the Griesmer bound. A special case of the bound yields both the extension of the Cadambe-Mazumdar bound and the Singleton-type bound for codes with locality $(r, {\delta})$, implying that the new bound is at least as good as these bounds. Furthermore, an upper bound on the asymptotic rate-distance tradeoff of LRCs is derived, and yields the tightest known upper bound for large relative minimum distances. Achievability results are also provided by deriving the locality of the family of Simplex codes together with a few examples of optimal codes.

The problem of private information retrieval (PIR) from coded storage systems with colluding, Byzantine, and unresponsive servers is considered. An explicit scheme using an [n, k] Reed-Solomon storage code is designed, protecting against t-collusion, and handling up to b Byzantine and r unresponsive servers, when n > k + t + 2b + r - 1. This scheme achieves a PIR rate of ((n - r - (k + 2b + t - 1))/n - r). In the case where the capacity is known, namely, when k = 1, it is asymptotically capacity achieving as the number of files grows. Finally, the scheme is adapted to symmetric PIR.

A private information retrieval (PIR) scheme on coded storage systems with colluding, byzantine, and non-responsive servers is presented. Furthermore, the scheme can also be used for symmetric PIR in the same setting. An explicit scheme using an [n, k] generalized Reed-Solomon storage code is designed, protecting against t-collusion and handling up to b byzantine and r non-responsive servers, when n ≥ n1'=(ν+1)k+t+2b+r-1, for some integer ν ≥ 1. This scheme achieves a PIR rate of 1-[(k+2b+t+r-1)/(n'-r)]. In the case where the capacity is known, namely when k=1, it is asymptotically capacity achieving as the number of files grows.

Recently, information-theoretic private information retrieval (PIR) from coded storage systems has gained a lot of attention, and a general star product PIR scheme was proposed. In this paper, the star product scheme is adopted, with appropriate modifications, to the case of private (e.g., video) streaming. It is assumed that the files to be streamed are stored on n servers in a coded form, and the streaming is carried out via a convolutional code. The star product scheme is defined for this special case, and various properties are analyzed for two channel models related to straggling and Byzantine servers, both in the baseline case as well as with colluding servers. The achieved PIR rates for the given models are derived and, for the cases where the capacity is known, the first model is shown to be asymptotically optimal, when the number of stripes in a file is large. The second scheme introduced in this work is shown to be the equivalent of block convolutional codes in the PIR setting. For the Byzantine server model, it is shown to outperform the trivial scheme of downloading stripes of the desired file separately without memory.

In Private Information Retrieval (PIR), one wants to download a file from a database without revealing to the database which file is being downloaded. Much attention has been paid to the case of the database being encoded across several servers, subsets of which can collude to attempt to deduce the requested file. With the goal of studying the achievable PIR rates in realistic scenarios, we generalize results for coded data from the case of all subsets of servers of size t colluding, to arbitrary subsets of the servers. We investigate the effectiveness of previous strategies in this new scenario, and present new results in the case where the servers are partitioned into disjoint colluding groups.

We consider a user-centric network-level coordination architecture for 5G heterogeneous Radio Access Networks (RANs), based on RAN softwarization and a centralized coordination framework. We describe the RAN as a set of logical RAN entities, related to cells in a Heterogeneous Network (HetNet), under the control of a central coordination entity. This description allows the creation of Network Functions (NFs) with an abstracted view of the network. We describe a centralized coordination framework, and then develop a NF for InterCell Interference Coordination (ICIC) in a 5G HetNet, optimizing the radio resource usage at network-level. We construct a Network Graph to abstract the problem of resource allocation and cell offloading, with the NF seeking for an optimal solution based on this abstraction. Simulations are performed in a HetNet scenario with a Tabu Search algorithm. Results show the feasibility of performing network-level coordination through a modular NF, with an abstracted view of the network.

Typically, locally repairable codes (LRCs) and regenerating codes have been studied independently of each other, and it has not been clear how the parameters of one relate to those of the other. In this paper, a novel connection between locally repairable codes and exact regenerating codes is established. Via this connection, locally repairable codes are interpreted as exact regenerating codes. Further, some of these codes are shown to perform better than time-sharing codes between minimum bandwidth regenerating and minimum storage regenerating codes.

Locally repairable codes (LRCs) are error correcting codes used in distributed data storage. Besides a global level, they enable errors to be corrected locally, reducing the need for communication between storage nodes. There is a close connection between almost affine LRCs and matroid theory which can be utilized to construct good LRCs and derive bounds on their performance. A generalized Singleton bound for linear LRCs with parameters (n; k; d; r; δ) was given in [N. Prakash et al., “Optimal Linear Codes with a Local-Error-Correction Property”, IEEE Int. Symp. Inf. Theory]. In this paper, a LRC achieving this bound is called perfect. Results on the existence and nonexistence of linear perfect (n; k; d; r; δ)-LRCs were given in [W. Song et al., “Optimal locally repairable codes”, IEEE J. Sel. Areas Comm.]. Using matroid theory, these existence and nonexistence results were later strengthened in [T. Westerbäck et al., “On the Combinatorics of Locally Repairable Codes”, Arxiv: 1501.00153], which also provided a general lower bound on the maximal achievable minimum distance dmax(n; k; r; δ) that a linear LRC with parameters (n; k; r; δ) can have. This article expands the class of parameters (n; k; d; r; δ) for which there exist perfect linear LRCs and improves the lower bound for dmax(n; k; r; δ). Further, this bound is proved to be optimal for the class of matroids that is used to derive the existence bounds of linear LRCs.

In this paper, locally repairable codes with all-symbol locality are studied. Methods to modify already existing codes are presented. It is also shown that, with high probability, a random matrix with a few extra columns guaranteeing the locality property is a generator matrix for a locally repairable code with a good minimum distance. The proof of the result provides a constructive method to find locally repairable codes. Finally, constructions of three infinite classes of optimal vector-linear locally repairable codes over a small alphabet independent of the code size are given.

This paper provides a link between matroid theory and locally repairable codes (LRCs) that are either linear or more generally almost affine. Using this link, new results on both LRCs and matroid theory are derived. The parameters (n, k, d, r, δ) of LRCs are generalized to matroids, and the matroid analog of the generalized singleton bound by Gopalan et al. for linear LRCs is given for matroids. It is shown that the given bound is not tight for certain classes of parameters, implying a nonexistence result for the corresponding locally repairable almost affine codes that are coined perfect in this paper. Constructions of classes of matroids with a large span of the parameters (n, k, d, r, δ) and the corresponding local repair sets are given. Using these matroid constructions, new LRCs are constructed with prescribed parameters. The existence results on linear LRCs and the nonexistence results on almost affine LRCs given in this paper strengthen the nonexistence and existence results on perfect linear LRCs given by Song et al.