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At least 19 recordsLinked to original sources
arXiv2012

On the Feasibility of Network Alignment for Three-Source Three-Destination Multiple Unicast Networks with Delays

A transform approach to network coding was introduced by Bavirisetti et al. (arXiv:1103.3882v3 [cs.IT]) as a tool to view wireline networks with delays as $k$-instantaneous networks (for some large $k$). When the local encoding kernels (LEKs) of the network are varied with every time block of length $k > 1$, the network is said to use block time varying LEKs. In this work, we propose a Precoding Based Network Alignment (PBNA) scheme based on transform approach and block time varying LEKs for three-source three-destination multiple unicast network with delays (3-S 3-D MUN-D). In a recent work, Meng et al. (arXiv:1202.3405v1 [cs.IT]) reduced the infinite set of sufficient conditions for feasibility of PBNA in a three-source three-destination instantaneous multiple unicast network as given by Das et al. (arXiv:1008.0235v1 [cs.IT]) to a finite set and also showed that the conditions are necessary. We show that the conditions of Meng et al. are also necessary and sufficient conditions for feasibility of PBNA based on transform approach and block time varying LEKs for 3-S 3-D MUN-D.

cs.IT↗
arXiv2018

On the Structure of Interlinked Cycle Structures with Interlocked Outer Cycles

For index coding problems with special structure on the side-information graphs called Interlinked Cycle (IC) structures index codes have been proposed in the literature (C. Thapa, L. Ong, and S. Johnson, "Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques", in IEEE Trans. Inf. Theory, vol. 63, no. 6, Jun. 2017, with a correction in "Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques", in arxiv (arxiv:1603.00092v2 [cs.IT] 25 Feb 2018)). Recently (S. Sasi and B.S. Rajan, "On Optimal Index Codes for Interlinked Cycle Structures with Outer Cycles," in arxiv (arXiv:1804.09120v1 [cs.IT]), 24 Apr 2018) for a generalization of IC structures called IC structures with interlocked outer cycles optimal length index codes have been reported and it is shown that the optimal length depends on the maximum number of disjoint outer cycles. In this paper we discuss certain structural properties of IC structures with interlocked outer cycles and provide a simple algorithm to find the maximum number of disjoint outer cycles.

cs.IT↗
arXiv2017

Simulation Study of Two Measures of Integrated Information

Background: Many authors have proposed Quantitative Theories of Consciousness (QTC) based on theoretical principles like information theory, Granger causality and complexity. Recently, Virmani and Nagaraj (arXiv:1608.08450v2 [cs.IT]) noted the similarity between Integrated Information and Compression-Complexity, and on this basis, proposed a novel measure of network complexity called Phi-Compression Complexity (Phi-C or $Φ^C$). Their computer simulations using Boolean networks showed that $Φ^C$ compares favorably to Giulio Tononi et al's Integrated Information measure $Φ$ 3.0 and exhibits desirable mathematical and computational characteristics. Methods: In the present work, $Φ^C$ was measured for two types of simulated networks: (A) Networks representing simple neuronal connectivity motifs (presented in Fig.9 of Tononi and Sporns, BMC Neuroscience 4(1), 2003); (B) random networks derived from Erdös-R ényi G(N, p)graphs. Code for all simulations was written in Python 3.6, and the library NetworkX was used to simulate the graphs. Results and discussions summary: In simulations A, for the same set of networks, $Φ^C$ values differ from the values of IIT 1.0 $Φ$ in a counter-intuitive manner. It appears that $Φ^C$ captures some invariant aspects of the interplay between information integration, network topology, graph composition and node entropy. While Virmani and Nagaraj (arXiv:1608.08450v2 [cs.IT]) sought to highlight the correlations between $Φ^C$ and IIT $Φ$, the results of simulations A highlight the differences between the two measures in the way they capture the integrated information. In simulations B, the results of simulations A are extended to the more general case of random networks. In the concluding section we outline the novel aspects of this paper, and our ongoing and future research.

q-bio.NC↗
arXiv2009

A Note on the Injection Distance

Koetter and Kschischang showed in [R. Koetter and F.R. Kschischang, "Coding for Errors and Erasures in Random Network Coding," IEEE Trans. Inform. Theory, {54(8), 2008] that the network coding counterpart of Gabidulin codes performs asymptotically optimal with respect to the subspace distance. Recently, Silva and Kschischang introduced in [D. Silva and F.R. Kschischang, "On Metrics for Error Correction in Network Coding," To appear in IEEE Trans. Inform. Theory, ArXiv: 0805.3824v4[cs.IT], 2009] the injection distance to give a detailed picture of what happens in noncoherent network coding. We show that the above codes are also asymptotically optimal with respect to this distance.

cs.IT↗
arXiv2015

Noisy Index Coding with Quadrature Amplitude Modulation (QAM)

This paper discusses noisy index coding problem over Gaussian broadcast channel. We propose a technique for mapping the index coded bits to M-QAM symbols such that the receivers whose side information satisfies certain conditions get coding gain, which we call the QAM side information coding gain. We compare this with the PSK side information coding gain, which was discussed in "Index Coded PSK Modulation," arXiv:1356200, [cs.IT] 19 September 2015.

cs.IT↗
arXiv2015

Locally Repairable Codes and Matroid Theory

Locally repairable codes (LRCs) are error correcting codes used in distributed data storage. A traditional approach is to look for codes which simultaneously maximize error tolerance and minimize storage space consumption. However, this tends to yield codes for which error correction requires an unrealistic amount of communication between storage nodes. LRCs solve this problem by allowing errors to be corrected locally. This thesis reviews previous results on the subject presented in [1]. These include that every almost affine LRC induces a matroid such that the essential properties of the code are determined by the matroid. Also, the generalized Singleton bound for LRCs can be extended to matroids as well. Then, matroid theory can be used to find classes of matroids that either achieve the bound, meaning they are optimal in a certain sense, or at least come close to the bound. This thesis presents an improvement to the results of [1] in both of these cases. [1] T. Westerbäck, R. Freij, T. Ernvall and C. Hollanti, "On the Combinatorics of Locally Repairable Codes via Matroid Theory", arXiv:1501.00153 [cs.IT], 2014.

cs.IT↗
arXiv2018

Index Codes for Interlinked Cycle Structures with Outer Cycles

Index code construction for a class of side-information graphs called interlinked cycle (IC) structures without outer cycles is given by Thapa, Ong and Johnson (C. Thapa, L. Ong, and S. Johnson, "Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques", in \textit{IEEE Trans. Inf. Theory, vol. 63, no. 6, Jun. 2017}, "Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques", in arxiv (arxiv:1603.00092v2 [cs.IT] 25 Feb 2018)). In this paper, construction of index codes for interlinked cycle (IC) structures with outer cycles is given along with a decoding algorithm.

cs.IT↗
arXiv2018

On Optimal Index Codes for Interlinked Cycle Structures with Outer Cycles

For index coding problems with special structure on the side-information graphs called Interlinked Cycle (IC) structures index codes have been proposed in the literature (C. Thapa, L. Ong, and S. Johnson, "Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques", in \textit{IEEE Trans. Inf. Theory, vol. 63, no. 6, Jun. 2017} with a correction in "Interlinked Cycles for Index Coding: Generalizing Cycles and Cliques", in arxiv (arxiv:1603.00092v2 [cs.IT] 25 Feb 2018)). In this paper we consider a generalization of IC structures called {\it IC structures with interlocked outer cycles}. For IC structures with interlocked outer cycles we show that the optimal length (also known as the minrank of the index coding problem) depends on the maximum number of disjoint outer cycles. We give two sufficient conditions such that if any of these is satisfied then we provide explicit optimal index code construction. The conditions mentioned above are shown to be not necessary by an explicit example.

cs.IT↗
arXiv2019

An Optimal Linear Error Correcting Delivery Scheme for Coded Caching with Shared Caches

Classical coded caching setting avails each user to have one dedicated cache. This is generalized to a more general shared cache scheme and the exact expression for the worst case rate was derived in [E. Parrinello, A. Unsal, P. Elia, "Fundamental Limits of Caching in Heterogeneous Networks with Uncoded Prefetching," available on arXiv:1811.06247 [cs.IT], Nov. 2018]. For this case, an optimal linear error correcting delivery scheme is proposed and an expression for the peak rate is established for the same. Furthermore, a new delivery scheme is proposed, which gives an improved rate for the case when the demands are not distinct.

cs.IT↗
arXiv2019

On Pliable Index Coding

A new variant of index coding problem termed as Pliable Index Coding Problem (PICOD) is formulated in [S. Brahma, C. Fragouli, "Pliable index coding", IEEE Transactions on Information Theory, vol. 61, no. 11, pp. 6192-6203, 2015]. In PICOD, we consider a server holding a set of messages and there is a set of clients having a subset of messages with them. Each client is satisfied if it receives any of the message which it doesn't have. We discuss about a class of PICOD where the side information is consecutive. We provide index codes for two extreme cases - for the class where each client gets exactly one desired message and for a class where total number of messages decoded by the effective clients is maximized. Another variant of index coding problem is - c-Constrained Pliable Index Coding Problem [Linqi Song, Christina Fragouli and Tianchu Zhao, "A Pliable Index Coding Approach to Data Shuffling," arXiv:1701.05540v3 [cs.IT] 3 May 2018]. It is basically PICOD with a c-constraint, i.e, each message is decoded by atmost c clients demanding that message. We provide index codes for some classes of this variant with consecutive side information.

cs.IT↗
arXiv2005

On the asymptotic accuracy of the union bound

A new lower bound on the error probability of maximum likelihood decoding of a binary code on a binary symmetric channel was proved in Barg and McGregor (2004, cs.IT/0407011). It was observed in that paper that this bound leads to a new region of code rates in which the random coding exponent is asymptotically tight, giving a new region in which the reliability of the BSC is known exactly. The present paper explains the relation of these results to the union bound on the error probability.

cs.IT↗
arXiv2012

On existence of Budaghyan-Carlet APN hexanomials

Budaghyan and Carlet constructed a family of almost perfect nonlinear (APN) hexanomials over a field with r^2 elements, and with terms of degrees r+1, s+1, rs+1, rs+r, rs+s, and r+s, where r = 2^m and s = 2^n with GCD(m,n)=1. The construction requires a technical condition, which was verified empirically in a finite number of examples. Bracken, Tan, and Tan (arXiv:1110.3177 [cs.it]) proved the condition holds when m = 2 or 4 (mod 6). In this article, we prove that the construction of Budaghyan and Carlet produces APN polynomials for all m and n. In the case where GCD(m,n) = k >= 1, Budaghyan and Carlet showed that the nonzero derivatives of the hexanomials are 2^k-to-one maps from F_{r^2} to F_{r^2}, provided the same technical condition holds. We prove their construction produces hexanomials with this differential property for all m and n.

math.CO↗
arXiv2018

On privacy amplification, lossy compression, and their duality to channel coding

We examine the task of privacy amplification from information-theoretic and coding-theoretic points of view. In the former, we give a one-shot characterization of the optimal rate of privacy amplification against classical adversaries in terms of the optimal type-II error in asymmetric hypothesis testing. This formulation can be easily computed to give finite-blocklength bounds and turns out to be equivalent to smooth min-entropy bounds by Renner and Wolf [Asiacrypt 2005] and Watanabe and Hayashi [ISIT 2013], as well as a bound in terms of the $E_γ$ divergence by Yang, Schaefer, and Poor [arXiv:1706.03866 [cs.IT]]. In the latter, we show that protocols for privacy amplification based on linear codes can be easily repurposed for channel simulation. Combined with known relations between channel simulation and lossy source coding, this implies that privacy amplification can be understood as a basic primitive for both channel simulation and lossy compression. Applied to symmetric channels or lossy compression settings, our construction leads to proto- cols of optimal rate in the asymptotic i.i.d. limit. Finally, appealing to the notion of channel duality recently detailed by us in [IEEE Trans. Info. Theory 64, 577 (2018)], we show that linear error-correcting codes for symmetric channels with quantum output can be transformed into linear lossy source coding schemes for classical variables arising from the dual channel. This explains a "curious duality" in these problems for the (self-dual) erasure channel observed by Martinian and Yedidia [Allerton 2003; arXiv:cs/0408008] and partly anticipates recent results on optimal lossy compression by polar and low-density generator matrix codes.

cs.IT↗
arXiv2020

Computing Similarity Queries for Correlated Gaussian Sources

Among many current data processing systems, the objectives are often not the reproduction of data, but to compute some answers based on the data resulting from queries. The similarity identification task is to identify the items in a database that are similar to a given query item for a given metric. The problem of compression for similarity identification has been studied in arXiv:1307.6609 [cs.IT]. Unlike classical compression problems, the focus is not on reconstructing the original data. Instead, the compression rate is determined by the desired reliability of the answers. Specifically, the information measure identification rate characterizes the minimum rate that can be achieved among all schemes which guarantee reliable answers with respect to a given similarity threshold. In this paper, we propose a component-based model for computing correlated similarity queries. The correlated signals are first decorrelated by the KLT transform. Then, the decorrelated signal is processed by a distinct D-admissible system for each component. We show that the component-based model equipped with KLT can perfectly represent the multivariate Gaussian similarity queries when optimal rate-similarity allocation applies. Hence, we can derive the identification rate of the multivariate Gaussian signals based on the component-based model. We then extend the result to general Gaussian sources with memory. We also study the models equipped with practical compone\nt systems. We use TC-$\triangle$ schemes that use type covering signatures and triangle-inequality decision rules as our component systems. We propose an iterative method to numerically approximate the minimum achievable rate of the TC-$\triangle$ scheme. We show that our component-based model equipped with TC-$\triangle$ schemes can achieve better performance than the TC-$\triangle$ scheme unaided on handling the multivariate Gaussian sources.

eess.SP↗
arXiv2023

General Distance Balancing for Quantum Locally Testable Codes

In this paper, we prove a lower bound on the soundness of quantum locally testable codes under the distance balancing construction of Evra et al. arXiv:2004.07935 [quant-ph]. Our technical contribution is that the new soundness of the quantum code is at least the old soundness divided by the classical code length (up to a constant factor). This allows us to use any classical code with independent checks when distance balancing, where previously only the repetition code had been considered for qLTCs. By using a good classical LDPC code, we are able to grow the dimension of the hypersphere product codes arXiv:1608.05089 [quant-ph] and the hemicubic codes arXiv:1911.03069 [quant-ph] while maintaining their distance and locality, but at the expense of soundness. From this, and also by distance balancing a chain complex of Cross et al. arXiv:2209.11405 [cs.IT], we obtain quantum locally testable codes of new parameters.

quant-ph↗
arXiv2021

Mathematical Theory of Computational Resolution Limit in Multi-dimensions

Resolving a linear combination of point sources from their band-limited Fourier data is a fundamental problem in imaging and signal processing. With the incomplete Fourier data and the inevitable noise in the measurement, there is a fundamental limit on the separation distance between point sources that can be resolved. This is the so-called resolution limit problem. Characterization of this resolution limit is still a long-standing puzzle despite the prevalent use of the classic Rayleigh limit. It is well-known that Rayleigh limit is heuristic and its drawbacks become prominent when dealing with data that is subjected to delicate processing, as is what modern computational imaging methods do. Therefore, more precise characterization of the resolution limit becomes increasingly necessary with the development of data processing methods. For this purpose, we developed a theory of "computational resolution limit" for both number detection and support recovery in one dimension in [arXiv:2003.02917[cs.IT], arXiv:1912.05430[eess.IV]]. In this paper, we extend the one-dimensional theory to multi-dimensions. More precisely, we define and quantitatively characterize the "computational resolution limit" for the number detection and support recovery problems in a general k-dimensional space. Our results indicate that there exists a phase transition phenomenon regarding to the super-resolution factor and the signal-to-noise ratio in each of the two recovery problems. Our main results are derived using a subspace projection strategy. Finally, to verify the theory, we proposed deterministic subspace projection based algorithms for the number detection and support recovery problems in dimension two and three. The numerical results confirm the phase transition phenomenon predicted by the theory.

eess.IV↗