Abstract

We present a hypergraph coloring based approach to pliable index coding (PICOD). We represent the given PICOD problem using a hypergraph consisting of m messages as vertices and the request-sets of the n clients as hyperedges. A conflict- free coloring of a hypergraph is an assignment of colors to its vertices so that each hyperedge contains a uniquely colored vertex. We show that various parameters arising out of conflict- free colorings (and some new variants) of the PICOD hypergraph result in new upper bounds for the optimal PICOD length. Using these new upper bounds, we show the existence of single-request PICOD schemes with length O(log2 Γ), where Γ is the maximum number of hyperedges overlapping with any hyperedge. For the t-request PICOD scenario, we show the existence of PICOD schemes of length max(O(log Γ log m), O(t log m)), under some mild conditions on the graph parameters. These results improve upon earlier work in general. We also show that our achievable lengths in the t-request case are asymptotically optimal, up to a multiplicative factor of log t. Our existence results are accompanied by randomized constructive algorithms, which have complexity polynomial in the parameters of the PICOD problem, in expectation or with high probability. Index Terms—Index Coding, Pliabl

Abstract

We study a family of subcodes of the $m$-dimensional product code $mathscr{C}^{otimes m}$ (`subproduct codes') that have a recursive Plotkin-like structure, and which include Reed-Muller (RM) codes and Dual Berman codes as special cases. We denote the codes in this family as $mathscr{C}^{otimes [r,m]}$, where mbox{$0 leq r leq m$} is the `order' of the code. These codes allow a `projection' operation that can be exploited in iterative decoding, viz., the sum of two carefully chosen subvectors of any codeword in $mathscr{C}^{otimes [r,m]}$ belongs to $mathscr{C}^{otimes [r-1,m-1]}$. Recursive subproduct codes provide a wide range of rates and block lengths compared to RM codes while possessing several of their structural properties, such as the Plotkin-like design, the projection property, and fast ML decoding of first-order codes. Our simulation results for first-order and second-order codes, that are based on a belief propagation decoder and a local graph search algorithm, show instances of subproduct codes that perform either better than or within $0.5$~dB of comparable RM codes and CRC-aided Polar codes.

Abstract

In shotgun sequencing, the input string (typically, a long DNA sequence composed of nucleotide bases) is sequenced as multiple overlapping fragments of much shorter lengths (called textit{reads}). Modelling the shotgun sequencing pipeline as a communication channel for DNA data storage, the capacity of this channel was identified in a recent work, assuming that the reads themselves are noiseless substrings of the original sequence. Modern shotgun sequencers however also output quality scores for each base read, indicating the confidence in its identification. Bases with low quality scores can be considered to be erased. Motivated by this, we consider the textit{shotgun sequencing channel with erasures}, where each symbol in any read can be independently erased with some probability $delta$. We identify achievable rates for this channel, using a random code construction and a decoder that uses typicality-like arguments to merge the reads.

Abstract

We study communication models for channels with erasures in which the erasure pattern can be controlled by an adversary with partial knowledge of the transmitted codeword. In particular, we design block codes for channels with binary inputs with an adversary who can erase a fraction p of the transmitted bits. We consider causal adversaries, who must choose to erase an input bit using knowledge of that bit and previously transmitted bits, and myopic adversaries, who can choose an erasure pattern based on observing the transmitted codeword through a binary erasure channel with random erasures. For both settings we design efficient (polynomial time) encoding and decoding algorithms that use randomization at the encoder only. Our constructions achieve capacity for the causal and "sufficiently myopic" models. For the "insufficiently myopic" adversary, the capacity is unknown, but existing converses show the capacity is zero for a range of parameters. For all parameters outside of that range, our construction achieves positive rates.

Abstract

We identify a family of binary codes whose structure is similar to Reed-Muller (RM) codes and which include RM codes as a strict subclass. The codes in this family are denoted as Cn(r,m) , and their duals are denoted as Bn(r,m) . The length of these codes is nm , where n≥2 , and r is their ‘order’. When n=2 , Cn(r,m) is the RM code of order r and length 2m . The special case of these codes corresponding to n being an odd prime was studied by Berman (1967) and Blackmore and Norton (2001). Following the terminology introduced by Blackmore and Norton, we refer to Bn(r,m) as the Berman code and Cn(r,m) as the dual Berman code. We identify these codes using a recursive Plotkin-like construction, and we show that these codes have a rich automorphism group, they are generated by the minimum weight codewords, and that they can be decoded up to half the minimum distance efficiently. Using a result of Kumar et al. (2016), we show that these codes achieve the capacity of the binary erasure channel (BEC) under bit-MAP decoding. Furthermore, except double transitivity, they satisfy all the code properties used by Reeves and Pfister to show that RM codes achieve the capacity of binary-input memoryless symmetric channels. Finally, when n is odd, we identify a large class of abelian codes that includes Bn(r,m) and Cn(r,m) and which achieves BEC capacity.

Abstract

We consider the standard broadcast setup with a single server broadcasting information to a number of clients, each of which contains local storage (called cache ) of some size, which can store some parts of the available files at the server. The centralized coded caching framework, consists of a caching phase and a delivery phase, both of which are carefully designed in order to use the cache and the channel together optimally. In prior literature, various combinatorial structures have been used to construct coded caching schemes. One of the chief drawbacks of many of these existing constructions is the large subpacketization level, which denotes the number of times a file should be split for the schemes to provide coding gain. In this work, using a new binary matrix model, we present several novel constructions for coded caching based on the various types of combinatorial designs and their q -analogs, which are also called subspace designs. While most of the schemes constructed in this work (based on existing designs) have a high cache requirement, they provide a rate that is either constant or decreasing, and moreover require competitively small levels of subpacketization, which is an extremely important feature in practical applications of coded caching. We also apply our constructions to the distributed computing framework of MapReduce, which consists of three phases, the Map phase, the Shuffle phase and the Reduce phase. Using our binary matrix framework, we present a new simple generic coded data shuffling scheme. Employing our designs-based constructions in conjunction with this new shuffling scheme, we obtain new coded computing schemes which have low file complexity, with marginally higher communication load compared to the optimal scheme for equivalent parameters. We show that our schemes can neatly extend to the scenario with full and partial stragglers also.

Abstract

We present a new class of private information retrieval (PIR) schemes that keep the identity of the file requested private in the presence of at most t colluding servers, based on the recent framework developed for such t-PIR schemes using star products of transitive codes. These t-PIR schemes employ the class of Berman codes as the storage-retrieval code pairs. Berman codes, which are binary linear codes of length nm for any n ≥ 2 and m ≥ 1 being positive integers, were recently shown to achieve the capacity of the binary erasure channel. We provide a complete characterization of the star products of the Berman code pairs, enabling us to calculate the PIR rate of the star product-based schemes that employ these codes. The schemes we present have flexibility in the number of servers, the PIR rate, the storage rate, and the collusion parameter t, owing to numerous codes available in the class of Berman codes. Due to space restrictions, the full version of this paper containing proofs of claims, additional examples, and results,

Abstract

A code of length n is said to be (combinatorially) (ρ, L)-list decodable if the Hamming ball of radius ρn around any vector in the ambient space does not contain more than L codewords. We study a recently introduced class of higher order MDS codes, which are closely related (via duality) to codes that achieve a generalized Singleton bound for list decodability. For some ℓ ≥ 1, higher order MDS codes of length n, dimension k, and order ℓ are denoted as (n, k)-MDS(ℓ) codes. We present a number of results on the structure of these codes, identifying the ‘extend-ability’ of their parameters in various scenarios. Specifically, for some parameter regimes, we identify conditions under which (n1, k1)-MDS(ℓ1) codes can be obtained from (n2, k2)-MDS(ℓ2) codes, via various techniques. We believe that these results will aid in efficient constructions of higher order MDS codes. We also obtain a new field size upper bound for the existence of such codes, which arguably improves over the best known existing bound, in some parameter regimes. Due to space restrictions, the full version of this paper containing proofs of claims is made available in [1].

Abstract

We consider the standard broadcast setup with a single server broadcasting information to a number of clients, each of which contains local storage (called cache) of some size, which can store some parts of the available files at the server. The centralized coded caching framework, consists of a caching phase and a delivery phase, both of which are carefully designed in order to use the cache and the channel together optimally. In prior literature, various combinatorial structures have been used to construct coded caching schemes. One of the chief drawbacks of many of these existing constructions is the large subpacketization level, which denotes the number of times a file should be split for the schemes to provide coding gain. In this work, using a new binary matrix model, we present several novel constructions for coded caching based on the various types of combinatorial designs and their q-analogs, which are also called subspace designs. While most of the schemes constructed in this work (based on existing designs) have a high cache requirement, they provide a rate that is either constant or decreasing, and moreover require competitively small levels of subpacketization, which is an extremely important feature in practical applications of coded caching. We also apply our constructions to the distributed computing framework of MapReduce, which consists of three phases, the Map phase, the Shuffle phase and the Reduce phase. Using our binary matrix framework, we present a new simple generic coded data shuffling scheme. Employing our designs-based constructions in conjunction with this new shuffling scheme, we obtain new coded computing schemes which have low file complexity, with marginally higher communication load compared to the optimal scheme for equivalent parameters. We show that our schemes can neatly extend to the scenario with full and partial stragglers also.

Abstract

We consider replication-based distributed storage systems in which each node stores the same quantum of data and each data bit stored has the same replication factor across the nodes. Such systems are referred to as balanced distributed databases. When existing nodes leave or new nodes are added to this system, the balanced nature of the database is lost, either due to the reduction in the replication factor, or the non-uniformity of the storage at the nodes. This triggers a rebalancing algorithm, that exchanges data between the nodes so that the balance of the database is reinstated. The goal is then to design rebalancing schemes with minimal communication load. In a recent work by Krishnan et al., coded transmissions were used to rebalance a carefully designed distributed database from a node removal or addition. These coded rebalancing schemes have optimal communication load, however, require the file-size to be at least exponential in the system parameters. In this work, we consider a cyclic balanced database (where data is cyclically placed in the system nodes) and present coded rebalancing schemes for node removal and addition in such a database. These databases (and the associated rebalancing schemes) require the file-size to be only cubic in the number of nodes in the system. We bound the advantage of our node removal rebalancing scheme over the uncoded scheme, and show that our scheme has a smaller communication load. In the node addition scenario, the rebalancing scheme presented is a simple uncoded scheme, which we show has optimal load.