V. Lalitha

Umberto Martínez-Peñas, V. Lalitha

In this work, we introduce maximally recoverable codes with locality and availability. We consider locally repairable codes (LRCs) where certain subsets of $ t $ symbols belong each to $ N $ local repair sets, which are pairwise disjoint after removing the $ t $ symbols, and which are of size $ r+δ-1 $ and can correct $ δ-1 $ erasures locally. Classical LRCs with $ N $ disjoint repair sets and LRCs with $ N $-availability are recovered when setting $ t = 1 $ and $ t=δ-1=1 $, respectively. Allowing $ t > 1 $ enables our codes to reduce the storage overhead for the same locality and availability. In this setting, we define maximally recoverable LRCs (MR-LRCs) as those that can correct any globally correctable erasure pattern given the locality and availability constraints. We then identify a large class of global erasure patterns that can be corrected by such MR-LRCs and prove that they are all the correctable patterns when $ t=1 $. We provide three explicit constructions of LRCs that can correct such erasure patterns (thus MR-LRCs for $ t=1 $), based on MSRD codes, each attaining the smallest finite-field sizes for some parameter regime. Finally, we extend the known lower bound on finite-field sizes from classical MR-LRCs to our setting (for any value of $ t $).

Charul Rajput, Mahak, V. Lalitha

We introduce generalized function-correcting partition codes (GFCPCs) that simultaneously protect multiple partitions of the message space against different numbers of errors. Given partitions with respective distance requirements, a GFCPC is a systematic encoding that guarantees, for each partition, a specified minimum Hamming distance between codewords whose messages lie in different blocks. This framework unifies and generalizes both function-correcting partition codes, which protect multiple functions with a common error-correction level, and function-correcting codes with data protection, which assign different levels of protection to data and a single function. We present a multi-step construction procedure for these codes and demonstrate it with some examples. We derive general upper and lower bounds on the optimal redundancy, including the upper bound which considers the join of different combinations of the partitions. We define the distance requirement matrix $\mathcal{D}$ for the GFCPCs and use it to characterize the optimal redundancy in terms of the shortest length of an associated $\mathcal{D}$-code. For two partitions of message space over the binary field, we establish improved lower bounds on the optimal redundancy under specific neighborhood conditions on the partitions. Through several examples, we demonstrate that the proposed framework can yield strictly smaller redundancy than both the sum of the individual FCPC redundancies and the redundancy of a single FCPC designed for the join partition with the highest distance (strongest protection required).

Penukonda Naga Chandana, Tejas Srivastava, Gowtham R. Kurri et al.

Dual discriminator generative adversarial networks (D2 GANs) were introduced to mitigate the problem of mode collapse in generative adversarial networks. In D2 GANs, two discriminators are employed alongside a generator: one discriminator rewards high scores for samples from the true data distribution, while the other favors samples from the generator. In this work, we first introduce dual discriminator $α$-GANs (D2 $α$-GANs), which combines the strengths of dual discriminators with the flexibility of a tunable loss function, $α$-loss. We further generalize this approach to arbitrary functions defined on positive reals, leading to a broader class of models we refer to as generalized dual discriminator generative adversarial networks. For each of these proposed models, we provide theoretical analysis and show that the associated min-max optimization reduces to the minimization of a linear combination of an $f$-divergence and a reverse $f$-divergence. This generalizes the known simplification for D2-GANs, where the objective reduces to a linear combination of the KL-divergence and the reverse KL-divergence. Finally, we perform experiments on 2D synthetic data and use multiple performance metrics to capture various advantages of our GANs.

Divija Swetha Gadiraju, V. Lalitha, Vaneet Aggarwal

Blockchain is a distributed ledger with wide applications. Due to the increasing storage requirement for blockchains, the computation can be afforded by only a few miners. Sharding has been proposed to scale blockchains so that storage and transaction efficiency of the blockchain improves at the cost of security guarantee. This paper aims to consider a new protocol, Secure-Repair-Blockchain (SRB), which aims to decrease the storage cost at the miners. In addition, SRB also decreases the bootstrapping cost, which allows for new miners to easily join a sharded blockchain. In order to reduce storage, coding-theoretic techniques are used in SRB. In order to decrease the amount of data that is transferred to the new node joining a shard, the concept of exact repair secure regenerating codes is used. The proposed blockchain protocol achieves lower storage than those that do not use coding, and achieves lower bootstrapping cost as compared to the different baselines.