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.

We consider a two-layer hierarchical coded caching network where a server with a library of$N$files is connected to$K_{1}$mirrors, each having a cache memory of size$M_{1}$files. Each mirror is further connected to$K_{2}$users, each equipped with a dedicated cache of size$M_{2}$files. In this paper, we propose two distinct coded caching schemes based on coded placement, corresponding to two distinct memory pairs, ($M_{1}, M_{2}$), which operate effectively in the high memory regime, i.e., when both$M_{1}$and$M_{2}$are large. We show that the proposed schemes outperform existing schemes at these memory points for smaller values of$K_{2}$. In setups where mirrors are positioned near each other, avoiding signal interference is crucial. This can be ensured by having all mirrors transmit using orthogonal carrier frequencies. To compare our schemes with existing ones, we used the composite rate metric, which accurately represents the total bandwidth utilized in such setups. The composite rate is given by$\bar{R}=R_{1}+K_{1} R_{2}$, where$R_{1}$is the rate from the server to the mirrors, and$R_{2}$is the rate from the mirrors to the users, with respect to$M_{1}$and$M_{2}$.

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 studies a two-layer hierarchical network in which some users are offline during the content delivery phase. A two-layer hierarchical network consists of a single server connected to multiple cache-aided mirror sites, and each mirror site is connected to a distinct set of cache-aided users. A scheme for such a hierarchical system with offline users has been proposed recently, but considered a special case where all mirror caches have zero memory, which is a significant limitation. We propose an array known as a hierarchical hotplug placement delivery array (HHPDA), which describes the placement and delivery phases of a coded caching scheme for a general two-layer hierarchical network with offline users. Further, we construct a class of HHPDAs using combinatorial t-designs.

In Pliable Private Information Retrieval (PPIR) with a single server, messages are partitioned into T non-overlapping classes. The user wants to retrieve a message from its desired class without revealing the identity of the desired class to the server. In [S. A. Obead, H. Y. Lin and E. Rosnes, “Single-Server Pliable Private Information Retrieval With Side Information,” arXiv:2305.06857 [cs.IT]], authors consider the problem of PPIR with Side Information (PPIR-SI), where the user now has side information. The user wants to retrieve any new message (not included in the side information) from its desired class without revealing the identity of the desired class. Identity of each message can be represented as a class-subclass index pair, where subclass index represents the membership of a message within a class. If the user does not know the subclass indices of its side information from a class, that class is termed as unidentifiable. Conversely, if the user knows the subclass indices of its side information from a class, that class is termed as identifiable. A scheme for the PPIR-SI is given by Obead et al. for the case when all classes are unidentifiable, i.e., the user is unaware of the subclass indices of all its side information, and this case is referred to as PPIR with Unidentifiable SI (PPIR-USI). In this paper, we study the problem of PPIR for the single server case when the side information is partially identifiable, and we term this case as PPIR with Identifiable Side Information (PPIR-ISI). There are η number of identifiable classes, where 1 ≤η≤r. We give a scheme for PPIR-ISI, and we prove that having some identifiable side information is advantageous by comparing the rate of the proposed scheme to the rate of the PPIR-USI scheme given by Obead et al. for some cases.

We consider the two-layered hierarchical coded caching problem introduced in [N. Karamchandani,$\mathbf{U}$. Niesen, M. A. Maddah-Ali, and S. N. Diggavi, “Hierarchical coded caching,” IEEE Trans. Inf. Theory, 2016], in which a server is connected to$K_{1}$mirrors, and each mirror is connected to$K_{2}$users. The mirrors and the users are equipped with the cache of size$M_{1}$and$M_{2}$, respectively. We propose a hierarchical coded caching scheme with coded placements that perform better than the existing schemes. In order to ensure a fair comparison with existing schemes, we introduce the notion of composite rate, defined as$\overline{R}=R_{1}+K_{1}R_{2}$, which consists of the rate from server to mirrors$R_{1}$and the rate from mirror to users$R_{2}$. The composite rate has not been discussed before in literature, and it represents the total consumed bandwidth in the system. Therefore, it is more appropriate to consider the composite rate along with$R_{1}$and$R_{2}$. For the proposed scheme, we show a trade-off between the global memory$\overline{M}=K_{1}M_{1}+K_{1}K_{2}M_{2}$of the system and the composite rate. We compare the proposed scheme with the existing hierarchical coded caching schemes using the proposed parameter “composite rate.”

We consider a hotplug coded caching systems in which some users are offline at the time of delivery. A placement delivery array (PDA) is a well-known tool for constructing a coded caching scheme for dedicated caches. In this paper, we introduce the concept of PDAs for hotplug coded caching schemes and refer to it as hotplug placement delivery array (HpPDA). We give an algorithm to describe the placement and the delivery phase of a hotplug coded caching scheme using HpPDA. We show that an existing hotplug coded caching scheme given in [Y. Ma and D. Tuninetti, “On coded caching systems with offline users,” in 2022 IEEE International Symposium on Information Theory (ISIT), 2022, pp. 1133–1138] corresponds to a class of HpPDAs, and then propose a method to further improve the rate of that scheme.

In the original coded caching model introduced by Maddah-Ali and Niesen in 2014, the server starts broadcasting only after it receives demands from all the users. So, all the users must be active during the delivery phase. In this work, we consider a coded caching model called hotplug coded caching in which some of the users are offline during the delivery phase. This model was first introduced by Ma and Tuninetti (“On Coded Caching Systems with Offline Users,” 2022 IEEE International Symposium on Information Theory). The concept of Hotplug Placement Delivery Arrays (HpPDAs) for the hotplug coded caching systems was introduced in (“Improved Hotplug Caching Schemes Using PDAs and t-Designs,” arXiv:2311.02856, 2024), in which the authors have constructed HpPDAs from t-designs. This work provides a new hotplug coded caching scheme from the existing HpPDAs. The performance comparison of the proposed scheme with the existing schemes is presented. When applied for HpPDAs from t-designs, our scheme outperforms the baseline scheme by Ma and Tuninetti, and the Improved t-scheme by Rajput and Rajan in some memory segments.

Caching is a method to ease the strain on the network during peak hours. Many caching approaches consider a single-layer system in which the server is connected to the users via an error-free shared link. However, in these systems, all the users involved in the placement phase must be present during the delivery phase. To address this, a hotplug model was presented, in which only some users may actually reveal their demand and be present during the delivery phase. On the other hand, a single-layered caching system was extended to the two-layered coded caching system. In this work, we consider a two-layered hierarchical system in which a server with$N$files is connected to$K_{1}$mirrors and each mirror is connected to$K_{2}$users. Out of all$K_{1}K_{2}$users, only$K^{\prime}$users are online during the delivery phase. All the mirrors and users are equipped with caches. We refer to this system as a hotplug hierarchical caching system. Here, we consider this system with the case when the cache memory of each mirror is zero, and propose a scheme using the concept of hotplug placement delivery arrays (HpPDAs).

Ong and Ho developed optimal linear index codes for single uniprior index coding problems (ICPs) by finding a spanning tree for each strongly connected component of their information-flow graphs, following which Thomas et al. considered the same class of ICPs over Rayleigh fading channels. They developed the min-max probability of error criterion for choosing an index code from the set of bandwidth-optimal linear index codes. Motivated by the above works, this paper deals with single uniprior ICPs over binary-input continuous-output channels. Minimizing the average probability of error is introduced as a criterion for further selection of index codes which is shown to be equivalent to minimizing the total number of transmissions used for decoding the message requests at all the receivers. An algorithm that generates a spanning tree with a lower value of this metric than the optimal star graph is also presented. A couple of lower bounds for the total number of transmissions, used by any optimal index code, are derived, and two classes of ICPs for which these bounds are tight are identified. An improvement of the proposed algorithm for information-flow graphs with bridges and a generalization of the improved algorithm for information-flow graphs obtainable as the union of strongly connected sub-graphs are presented, and some optimality results are derived.

Ong and Ho developed optimal linear index codes for single uniprior index coding problems (ICPs) by finding a spanning tree for each of the strongly connected components of the corresponding information-flow graphs, following which Thomas et al. considered the same class of ICPs over Rayleigh fading channel. They developed the min-max probability of error criterion for choosing an index code which minimized the probability of error at the receivers and showed that there always exist optimal linear index codes for which any receiver takes at most two transmissions to decode a requested message. Motivated by the above works, this paper considers single uniprior ICPs over Rayleigh fading channels for which minimizing average probability of error is shown to be a criterion for further selection of index codes. The optimal index code w.r.t this criterion is shown to be one that minimizes the total number of transmissions used for decoding the message requests at all the receivers. An algorithm that generates a spanning tree which has a lower value of this metric as compared to the optimal star graph is also presented. For a given set of parameters of single uniprior ICPs, a lower bound for the total number of transmissions used by any optimal index code is derived, and a class of ICPs for which this bound is tight is identified.

In this paper, we propose a scheme for the problem of cache-aided multi-user private information retrieval with small caches. All users want to retrieve a file without revealing their demands to the databases. During off-peak hours, all the users will fill their caches, and when required, users will demand their desired files by cooperatively generating query sets for each database. After receiving the transmissions from databases, all the users should get their desired files using transmitted data and their cache contents. This problem has been studied in [X. Zhang, K. Wan, H. Sun, M. Ji and G. Caire, "Fundamental limits of cache-aided multiuser private information retrieval", IEEE Trans. Commun., 2021], in which authors proposed a product design scheme. In this paper, we propose a scheme that gives a better rate for a small value of cache size than the product design scheme. We consider a slightly different approach for the placement phase. Instead of a database filling the caches of all users directly, a database will broadcast cache content for all users on a shared link, and then the users will decide together which part of the broadcasted content will be stored in the cache of each user. This variation facilitates maintaining the privacy constraint at a reduced rate.

Coded caching techniques help to reduce the traffic overload on the server during peak-traffic hours. Most of the existing schemes consider all transmissions to be over noiseless channels, whereas noise is inherent in wireless communication. In this paper, the multi-access coded caching scheme proposed in the paper [“Maddah-Ali-Niesen Scheme for Multi-access Coded Caching,” ITW 2021], is studied when the server-user shared link and all the cache-user links are noisy. This coded caching has been shown to be information-theoretically optimal in [“Fundamental Limits of Combinatorial Multi-Access Caching,” IEEE Transactions on Information Theory, Feb. 2023]. For binary modulated transmissions, the probability that a bit of a requested file is decoded in error at a user is derived when the transmissions are over binary-symmetric, AWGN, and Rayleigh fading channels. Further, the effect of varying the cache access degree (which is the number of caches accessed by each user), and the cache memory size on the probability of bit error performance at a user are also analyzed. Simulation results validating the findings in this paper are also presented.

In this paper, we study a class of locally recoverable codes in which any two recovering sets of a coordinate position can intersect at a small number of elements, but the intersection of any three recovering sets of a coordinate is empty. This subclass of LRC codes with intersecting recovering sets is important because a code with t recovering sets in this subclass can handle $\left\lceil {\frac{t}{2}} \right\rceil$ simultaneous erasures locally while having all the properties of the parent class (LRC codes with intersecting recovering sets).