Scalar Linear Codes Research Articles

Information Leakage in Index Coding With Sensitive and Nonsensitive Messages

Index coding can be viewed as a compression problem with multiple decoders with side information. In such a setup, an encoder compresses a number of messages into a common codeword such that every decoder can decode its requested messages with the help of knowing some other messages as side information. In this paper, we study how much information is leaked to a guessing adversary observing the codeword in index coding, where some messages in the system are sensitive and others are not. The non-sensitive messages can be used by the encoder in a manner similar to secret keys to mitigate leakage of the sensitive messages to the adversary. We first characterize the optimal information leakage rate of a given index coding problem by the optimal compression rate of a related problem, which is constructed by adding an extra decoder with certain parameters to the original problem. Both the achievability and converse of the characterization are derived from a graph-theoretic perspective based on confusion graphs (Alon et al. 2008). In particular, the achievable coding scheme is a randomized mapping exploiting certain symmetrical properties of the confusion graph. Our second main result is a practical deterministic linear coding scheme, developed from the rank minimization method based on fitting matrices (Bar-Yossef et al. 2011). The linear scheme leads to an upper bound on the optimal leakage rate, which is proved to be tight over all deterministic scalar linear codes. While it is shown through an example that simultaneously achieving optimal compression and leakage rates is not always possible, time-sharing between different schemes could be used to balance the compression and leakage rates. Finally, we show how our results can be applied to different variants of index coding.

On Rack-Aware Cooperative Regenerating Codes and Epsilon-MSCR Codes

In distributed storage systems, cooperative regenerating codes tradeoff storage for repair bandwidth in the case of multiple node failures. In rack-aware distributed storage systems, there is no cost associated with transferring symbols within a rack. Hence, the repair bandwidth will only take into account cross-rack transfer. Rack-aware regenerating codes for the case of single node failures have been studied and their repair bandwidth tradeoff characterized. In this paper, we consider the framework of rack-aware cooperative regenerating codes for the case of multiple node failures where the node failures are uniformly distributed among a certain number of racks. We characterize the storage repair-bandwidth tradeoff as well as derive the minimum storage and minimum repair bandwidth points of the tradeoff. We also provide explicit constructions of minimum bandwidth rack-aware cooperative regenerating codes and minimum storage rack-aware cooperative regenerating codes for a large range of parameters. We also consider another extension of minimum storage cooperative regenerating codes, in which we design slightly sub-optimal cooperative regenerating codes with much lower sub-packetization. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> -MSR codes are a class of codes introduced to tradeoff sub-packetization level for a slight increase in the repair bandwidth for the case of single node failures. We introduce the framework of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> -MSCR codes which allow for a similar tradeoff for the case of multiple node failures. We present a construction of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> -MSCR codes, which can recover from two node failures, by concatenating a class of MSCR codes and scalar linear codes. We give a repair procedure to repair the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> -MSCR codes in the event of two node failures and calculate the repair bandwidth for the same. We characterize the increase in repair bandwidth incurred by the method in comparison with the optimal repair bandwidth. Finally, we show the subpacketization level of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> -MSCR codes scales logarithmically in the number of nodes.

Permute and Add Network Codes via Group Algebras

A class of network codes have been proposed in the literature where the symbols transmitted on network edges are binary vectors and the coding operation performed in network nodes consists of the application of (possibly several) permutations on each incoming vector and XOR-ing the results to obtain the outgoing vector. These network codes, which we will refer to as <i>permute-and-add</i> network codes, involve simpler operations and are known to provide lower complexity solutions than scalar linear network codes. The complexity of these codes is determined by their <i>degree</i> which is the number of permutations applied on each incoming vector to compute an outgoing vector. Constructions of permute-and-add network codes for multicast networks are known. In this paper, we provide a new framework based on group algebras to design permute-and-add network codes for arbitrary (not necessarily multicast) networks. Our framework allows the use of any finite group of permutations (including circular shifts, proposed in prior works) and admits a trade-off between coding rate and the degree of the code. Further, our technique permits elegant recovery and generalizations of the key results on permute-and-add network codes known in the literature.

Private Index Coding

We study the fundamental problem of index coding under an additional privacy constraint that requires each receiver to learn nothing more about the collection of messages beyond its demanded messages from the server and what is available to it as side information. To enable such private communication, we allow the use of a collection of independent secret keys, each of which is shared amongst a subset of users and is known to the server. The goal is to study properties of the key access structures that make the problem feasible and then design encoding and decoding schemes efficient in the size of the server transmission as well as the sizes of the secret keys. We call this the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">private index coding</i> problem. We begin by characterizing the key access structures that make private index coding feasible. We also give conditions to check if a given linear scheme is a valid private index code. For up to three users, we characterize the rate region of feasible server transmission and key rates, and show that all feasible rates can be achieved using scalar linear coding and time sharing; we also show that scalar linear codes are sub-optimal for four receivers. The outer bounds used in the case of three users are extended to arbitrary number of users and seen as a generalized version of the well-known polymatroidal bounds for the standard non-private index coding. We also show that the presence of common randomness and private randomness does not change the rate region. Furthermore, we study the case where the server has the ability to multicast to any subset of users, and demonstrate how this flexibility can be used to provide privacy and characterize the minimum number of server multicasts required.

An Update-Based Maximum Column Distance Coding Scheme for Index Coding

In this paper, we propose a new scalar linear coding scheme for the index coding problem called update-based maximum column distance (UMCD) coding scheme. The central idea in each transmission is to code messages such that one of the receivers with the minimum size of side information is instantaneously eliminated from unsatisfied receivers. One main contribution of the paper is to prove that the other satisfied receivers can be identified after each transmission, using a polynomial-time algorithm solving the well-known maximum cardinality matching problem in graph theory. This leads to determining the total number of transmissions without knowing the coding coefficients. Once this number and what messages to transmit in each round are found, we then propose a method to determine all coding coefficients from a sufficiently large finite field. We provide concrete instances where the proposed UMCD coding scheme has a better broadcast performance compared to the most efficient existing coding schemes, including the recursive scheme (Arbabjolfaei and Kim, 2014) and the interlinked-cycle cover (ICC) scheme (Thapa <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> , 2017). We prove that the proposed UMCD coding scheme performs at least as well as the MDS coding scheme in terms of broadcast rate. By characterizing two classes of index coding instances, we show that the gap between the broadcast rates of the recursive and ICC schemes and the UMCD scheme grows linearly with the number of messages. Then, we extend the UMCD coding scheme to its vector version by applying it as a basic coding block to solve the subinstances.

Locally Decodable Index Codes

An index code for broadcast channel with receiver side information is locally decodable if each receiver can decode its demand by observing only a subset of the transmitted codeword symbols instead of the entire codeword. Local decodability in index coding is known to reduce receiver complexity, improve user privacy and decrease decoding error probability in wireless fading channels. Conventional index coding solutions assume that the receivers observe the entire codeword, and as a result, for these codes the number of codeword symbols queried by a user per decoded message symbol, which we refer to as locality , could be large. In this paper, we pose the index coding problem as that of minimizing the broadcast rate for a given value of locality (or vice versa) and designing codes that achieve the optimal trade-off between locality and rate. We identify the optimal broadcast rate corresponding to the minimum possible value of locality for all single unicast problems. We present new structural properties of index codes which allow us to characterize the optimal trade-off achieved by: vector linear codes when the side information graph is a directed cycle; and scalar linear codes when the minrank of the side information graph is one less than the order of the problem. We also identify the optimal trade-off among all codes, including non-linear codes, when the side information graph is a directed 3-cycle. Finally, we present techniques to design locally decodable index codes for arbitrary single unicast problems and arbitrary values of locality.

User-Cache Aided Transmission With Index Coding in $K$ -User Downlink Channels

This paper considers a single-input single-output broadcast channel with receiver side information. We find the optimal clique cover index code and the optimal transmission time allocation that minimize outage probability when total transmission time is limited. As our problem is NP-hard, we first find the optimal time allocation for a given index code. Then, we describe a brute-force algorithm that finds the set of all decodable index codes and chooses the optimal one adopting the optimal time allocation. To reduce the computational complexity of the brute-force algorithm, we propose a pruning algorithm which solves the same problem using the Hasse diagram but does not harm the optimality. Our analysis reveals that the optimal index code is dependent on the channel conditions, not simply on the number of required transmissions, which implies that the index coding-channel coupling improves the outage performance. It is also shown that this claim is still valid for general scalar linear index coding. Our simulation results verify that our proposed schemes effectively reduce the outage probability compared to other reference schemes, and our pruning algorithm considerably reduces the computational complexity required for the brute-force algorithm.

Grassmannian Codes With New Distance Measures for Network Coding

Grassmannian codes are known to be useful in error correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet size. The multicast networks which were used for this purpose are generalized combination networks. In both the scalar and the vector network coding solutions, the subspace distance is used as the distance measure for the codes which solve the network coding problem in the generalized combination networks. In this paper, we show that the subspace distance can be replaced with two other possible distance measures which generalize the subspace distance. These two distance measures are shown to be equivalent under an orthogonal transformation. It is proved that the Grassmannian codes with the new distance measures generalize the Grassmannian codes with the subspace distance and the subspace designs with the strength of the design. Furthermore, optimal Grassmannian codes with the new distance measures have minimal requirements for the network coding solutions of some generalized combination networks. The coding problems related to these two distance measures, especially with respect to network coding, are discussed. Finally, by using these new concepts, it is proved that the codes in the Hamming scheme form a subfamily of the Grassmannian codes.

On Encoding and Decoding of Circular-Shift Linear Network Codes

In a multicast network, a circular-shift linear code can be induced from a scalar linear code such that equipped with a nontrivially constructed source encoding matrix, and the induced code qualifies to be a circular-shift linear solution at a certain rate. However, for such an induced solution, there was no explicit characterization of the corresponding decoding matrix at every receiver. In this letter, we formulate a general rule to construct a source encoding matrix and a corresponding decoding matrix at every receiver. By application of the general rule, we further design new instances of the encoding and the decoding matrices, which can be expressed in a systematic form and involve fewer binary operations.

Optimal Index Codes for a New Class of Interlinked Cycle Structure

The interlinked cycle (IC) structure that generalizes cycles and cliques was defined by Thapa, Ong, and Johnson. Interlinked-cycle-cover (ICC) scheme that leverages IC structures in digraphs to construct scalar linear index codes was proposed. A class of infinitely many digraphs, where the proposed scalar linear codes based on ICC scheme are optimal were characterized by Thapa et. al . In this paper, we provide an addition to this class by providing optimal length index codes for IC structures with one cycle among non-inner vertex set.