In this paper we study error-correcting codes for the storage of data in synthetic deoxyribonucleic acid (DNA). We investigate a storage model where a data set is represented by an unordered set of $M$ sequences, each of length $L$ . Errors within that model are a loss of whole sequences and point errors inside the sequences, such as insertions, deletions and substitutions. We derive Gilbert-Varshamov lower bounds and sphere packing upper bounds on achievable cardinalities of error-correcting codes within this storage model. We further propose explicit code constructions than can correct errors in such a storage system that can be encoded and decoded efficiently. Comparing the sizes of these codes to the upper bounds, we show that many of the constructions are close to optimal.

List decoding of insertions and deletions in the Levenshtein metric is considered. The Levenshtein distance between two sequences is the minimum number of insertions and deletions needed to turn one of the sequences into the other. In this paper, a Johnson-like upper bound on the maximum list size when list decoding in the Levenshtein metric is derived. This bound depends only on the length and minimum Levenshtein distance of the code, the length of the received word, and the alphabet size. It shows that polynomial-time list decoding beyond half the Levenshtein distance is possible for many parameters. Further, we also prove a lower bound on list decoding of deletions with the well-known binary Varshamov-Tenengolts codes, which shows that the maximum list size grows exponentially with the number of deletions. Finally, an efficient list decoding algorithm for two insertions/deletions with VT codes is given. This decoder can be modified to a polynomial-time list decoder of any constant number of insertions/deletions.

This paper considers vector network coding solutions based on rank-metric codes and subspace codes. The main result of this paper is that vector solutions can significantly reduce the required alphabet size compared to the optimal scalar linear solution for the same multicast network. The multicast networks considered in this paper have one source with h messages, and the vector solution is over a field of size q with vectors of length t. For a given network, let the smallest field size for which the network has a scalar linear solution be qs, then the gap in the alphabet size between the vector solution and the scalar linear solution is defined to be qs–qt. In this contribution, the achieved gap is q(h−2)t2/h+o(t) for any q ≥ 2 and any even h ≥ 4. If h ≥ 5 is odd, then the achieved gap of the alphabet size is q(h−3)t2/(h−1)+o(t). Previously, only a gap of size size one had been shown for networks with a very large number of messages. These results imply the same gap of the alphabet size between the optimal scalar linear and some scalar nonlinear network coding solution for multicast networks. For three messages, we also show an advantage of vector network coding, while for two messages the problem remains open. Several networks are considered, all of them are generalizations and modifications of the well-known combination networks. The vector network codes that are used as solutions for those networks are based on subspace codes, particularly subspace codes obtained from rank-metric codes. Some of these codes form a new family of subspace codes, which poses a new research problem.

Federated learning (FL) has emerged as a promising approach for collaboratively training machine learning models while preserving data privacy. Due to its decentralized nature, FL is vulnerable to poisoning attacks, where malicious clients compromise the global model through altered data or updates. Identifying such malicious clients is crucial for ensuring the integrity of FL systems. This task becomes particularly challenging under privacy-enhancing protocols such as secure aggregation, creating a fundamental trade-off between privacy and security. In this work, we propose FedGT, a novel framework designed to identify malicious clients in FL with secure aggregation while preserving privacy. Drawing inspiration from group testing, FedGT leverages overlapping groups of clients to identify the presence of malicious clients via a decoding operation. The clients identified as malicious are then removed from the model training, which is performed over the remaining clients. By choosing the size, number, and overlap between groups, FedGT strikes a balance between privacy and security. Specifically, the server learns the aggregated model of the clients in each group—vanilla federated learning and secure aggregation correspond to the extreme cases of FedGT with group size equal to one and the total number of clients, respectively. The effectiveness of FedGT is demonstrated through extensive experiments on three datasets in a cross-silo setting under different data-poisoning attacks. These experiments showcase FedGT’s ability to identify malicious clients, resulting in high model utility. We further show that FedGT significantly outperforms the private robust aggregation approach based on the geometric median recently proposed by Pillutla et al. and the robust aggregation technique Multi-Krum in multiple settings.

Decoding sequences that stem from multiple transmissions of a codeword over an insertion, deletion, and substitution channel is a critical component of efficient deoxyribonucleic acid (DNA) data storage systems. In this paper, we consider a concatenated coding scheme with an outer nonbinary low-density parity-check code or a polar code and either an inner convolutional code or a time-varying block code. We propose two novel decoding algorithms for inference from multiple received sequences, both combining the inner code and channel to a joint hidden Markov model to infer symbolwise a posteriori probabilities (APPs). The first decoder computes the exact APPs by jointly decoding the received sequences, whereas the second decoder approximates the APPs by combining the results of separately decoded received sequences and has a complexity that is linear with the number of sequences. Using the proposed algorithms, we evaluate the performance of decoding multiple received sequences by means of achievable information rates and Monte-Carlo simulations. We show significant performance gains compared to a single received sequence. In addition, we succeed in improving the performance of the aforementioned coding scheme by optimizing both the inner and outer codes.