Optimal Achievable Rates for Computation With Random Homologous Codes

Sen, Pinar; Lim, Sung Hoon; Kim, Young-Han

doi:10.1109/isit.2018.8437750

Cited by 5 publications

(4 citation statements)

References 28 publications

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“…Another important aspect of our framework is the resulting simultaneous joint decoding rate region for computeforward. The joint typicality decoder presented in this paper was shown to be optimal with respect to nested linear codes in [48] for the K = 2, L = 1 case. On the other hand, the sharpest-known analysis for a lattice-based compute-forward strategy relies on suboptimal sequential decoding [38] due to the technical limitations in analyzing joint decoders for lattice codes [49].…”

Section: Discussionmentioning

confidence: 90%

See 1 more Smart Citation

Compute-Forward for DMCs: Simultaneous Decoding of Multiple Combinations

Lim

Feng

Pastore

et al. 2020

IEEE Trans. Inform. Theory

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Consider a receiver in a multiuser network that wishes to decode several messages. Simultaneous joint typicality decoding is one of the most powerful techniques for determining the fundamental limits at which reliable decoding is possible. This technique has historically been used in conjunction with random i.i.d. codebooks to establish achievable rate regions for networks. Recently, it has been shown that, in certain scenarios, nested linear codebooks in conjunction with "single-user" or sequential decoding can yield better achievable rates. For instance, the compute-forward problem examines the scenario of recovering L ≤ K linear combinations of transmitted codewords over a K-user multipleaccess channel (MAC), and it is well established that linear codebooks can yield higher rates. Here, we develop bounds for simultaneous joint typicality decoding used in conjunction with nested linear codebooks, and apply them to obtain a larger achievable region for compute-forward over a K-user discrete memoryless MAC. The key technical challenge is that competing codeword tuples that are linearly dependent on the true codeword tuple introduce statistical dependencies, which requires careful partitioning of the associated error events. Index Terms Compute-forward, joint decoding, linear codes, multiple-access channel I. INTRODUCTION For several decades, decode-forward [1], compress-forward [1], and amplify-forward [2] have served as the fundamental building blocks of transmission strategies for relay networks. These three relaying strategies were initially developed on canonical network models such as the relay channel and diamond relay network using random

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Section: Discussionmentioning

confidence: 90%

“…The last two rate inequalities (47)- (48) are combined via a logical 'or' (due to the union over S). Recombining the above three case distinctions on the coefficient pair (c 1 , c 2 ) via a logical 'and' (due to union over C) yields the rate region R LMAC as defined in (10).…”

Section: Appendix a Proof Of Corollarymentioning

confidence: 99%

Compute-Forward for DMCs: Simultaneous Decoding of Multiple Combinations

Lim

Feng

Pastore

et al. 2020

IEEE Trans. Inform. Theory

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“…, span(A) ⊆ span(B), we have a bound on P (E 3 ∩ E c 1 |M) that tends to zero as n → ∞ if for all full rank C ∈ F L C ×L B q , 0 ≤ L C < L B , there exists an S that satisfies (13) and…”

Section: P(e 3 ∩ E Cmentioning

confidence: 99%

Towards an Algebraic Network Information Theory: Distributed Lossy Computation of Linear Functions

Lim¹,

Feng

Pastore

et al. 2019

2019 IEEE International Symposium on Information Theory (ISIT)

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Consider the important special case of the K-user distributed source coding problem where the decoder only wishes to recover one or more linear combinations of the sources. The work of Körner and Marton demonstrated that, in some cases, the optimal rate region is attained by random linear codes, and strictly improves upon the best-known achievable rate region established via random i.i.d. codes. Recent efforts have sought to develop a framework for characterizing the achievable rate region for nested linear codes via joint typicality encoding and decoding. Here, we make further progress along this direction by proposing an achievable rate region for simultaneous joint typicality decoding of nested linear codes. Our approach generalizes the results of Körner and Marton to computing an arbitrary number of linear combinations and to the lossy computation setting.

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“…Remark 4: The achievable rate region in Proposition 4 is the largest region thus far established in the literature. As a matter of fact, there is some indication that this region is optimal in the sense that it cannot be improved by using maximum likelihood decoding [9], [10]. Still, it is in general strictly smaller than the capacity region of the k-sender DM-MAC.…”

Section: A Shapingmentioning

confidence: 99%

Homologous codes for multiple access channels

Sen

Kim

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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Building on recent development by Padakandla and Pradhan, and by Lim, Feng, Pastore, Nazer, and Gastpar, this paper studies the potential of structured nested coset coding as a complete replacement for random coding in network information theory. The roles of two techniques used in nested coset coding to generate nonuniform codewords, namely, shaping and channel transformation, are clarified and illustrated via the simple example of the two-sender multiple access channel. While individually deficient, the optimal combination of shaping and channel transformation is shown to achieve the same performance as traditional random codes for the general two-sender multiple access channel. The achievability proof of the capacity region is extended to the multiple access channels with more than two senders, and with one or more receivers. A quantization argument consistent with the construction of nested coset codes is presented to prove achievability for their Gaussian counterparts. These results open up new possibilities of utilizing nested coset codes with the same generator matrix for a broader class of applications. I. INTRODUCTIONRandom independently and identically distributed (i.i.d.) code ensembles play a fundamental role in network information theory, with most existing coding schemes built on them; see, for example, [1]- [3]. As shown by the classical example by Körner and Marton [4], however, using the same code at multiple users can achieve strictly better performance for some communication problems. Recent studies illustrate the benefit of such structured coding for computing linear combinations in [5]-[10], for the interference channels in [11]-[14], and for the multiple access channels with state information in [15]. Consequently, there has been a flurry of research activities on structured coding in network information theory, facilitated in part by several standalone workshops and tutorials at major conferences by leading researchers.Most of the existing results are based on lattice codes or linear codes on finite alphabets. Recently, Padakandla and Pradhan [15] brought a new dimension to the arsenal of structured coding by developing nested coset codes for network information theory; see also Miyake [16] for nested coset codes for point-to-point communication. In these nested coset coding schemes, a coset code of a rate higher than the target is first generated randomly. A codeword of a desired property (such as type or joint type) is then selected from a subset (a coset of a subcode). This construction is reminiscent of the multicoding scheme in Gelfand-Pinsker coding for channels with state and Marton coding for broadcast channels. But in a sense, nested coset coding is more fundamental in that the scheme at its core is relevant even for single-user communication. By a careful combination of individual and common parts of coset codes, the proposed coding scheme in [15] achieves rates for multiple access channels (MACs) with state beyond what can be achieved by existing random or structured coding sche...

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Optimal Achievable Rates for Computation With Random Homologous Codes

Cited by 5 publications

References 28 publications

Compute-Forward for DMCs: Simultaneous Decoding of Multiple Combinations

Compute-Forward for DMCs: Simultaneous Decoding of Multiple Combinations

Towards an Algebraic Network Information Theory: Distributed Lossy Computation of Linear Functions

Homologous codes for multiple access channels

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