MCMC methods for entropy optimization and nonlinear network coding

Given n discrete random variables, its entropy vector is the 2 n − 1 dimensional vector obtained from the joint entropies of all non-empty subsets of the random variables. It is well known that there is a one-toone correspondence between such an entropy vector and a certain group-characterizable vector obtained from a finite group and n of its subgroups [3]. This correspondence may be useful for characterizing the space of entropic vectors and for designing network codes. If one restricts attention to abelian groups then not all entropy vectors can be obtained. This is an explanation for the fact shown by Dougherty et al[4] that linear network codes cannot achieve capacity in general network coding problems (since linear network codes form an abelian group). All abelian group-characterizable vectors, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called the Ingleton inequality. It is therefore of interest to identify groups that violate the Ingleton inequality. In this paper, we study the problem of finding nonabelian finite groups that yield characterizable vectors which violate the Ingleton inequality. Using a refined computer search, we find the symmetric group S 5 to be the smallest group that violates the Ingleton inequality. Careful study of the structure of this group, and its subgroups, reveals that it belongs to the Ingleton-violating family P GL(2, q) with a prime power q ≥ 5, i.e., the projective group of 2 × 2 nonsingular matrices with entries in F q . We further interpret this family of groups, and their subgroups, using the theory of group actions and identify the subgroups as certain stabilizers. We also extend the construction to more general groups such as P GL(n, q) and GL(n, q). The families of groups identified here are therefore good candidates for constructing network codes more powerful than linear network codes, and we discuss some considerations for constructing such group network codes. Index TermsPortions of this work were presented at the

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“…A perhaps more geometrically meaningful scaling factor is the 2-norm of the entropy vector, as proposed in [32]:…”

Section: Discussionmentioning

confidence: 99%

On Ingleton-Violating Finite Groups

Mao

Thill

Hassibi

2017

IEEE Trans. Inform. Theory

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“…Most network coding applications are developed with linear network coding, although non-linear network coding has also been studied, e.g., in Dougherty et al [26], Kosut et al [72], Lehman and Lehaman [79], Li et al [83], Shadbakht and Hassibi [111]. In linear network coding, packets transferred through a network are viewed as symbols in GF(q), and arithmetic operations of those are defined on GF(q).…”

Section: Linear Network Codingmentioning

confidence: 99%

Survey of Network Coding and Its Applications

Matsuda

Noguchi

Takine

2011

IEICE Trans. Commun.

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SUMMARYThis survey summarizes the state-of-the-art research on network coding, mainly focusing on its applications to computer networking. Network coding generalizes traditional store-and-forward routing techniques by allowing intermediate nodes in networks to encode several received packets into a single coded packet before forwarding. Network coding was proposed in 2000, and since then, it has been studied extensively in the field of computer networking. In this survey, we first summarize linear network coding and provide a taxonomy of network coding research, i.e., the network coding design problem and network coding applications. Moreover, the latter is subdivided into throughput/capacity enhancement, robustness enhancement, network tomography, and security. We then discuss the fundamental characteristics of network coding and diverse applications of network coding in details, following the above taxonomy.

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“…The zeroerror capacity of a class of networks was characterized in [7] by using quasi-uniform network codes. An explicit non-linear quasi-uniform code was presented in [8], [9] for the wellknown Vámos network. Thus, quasi-uniform random vectors are very important in code design.…”

Section: Introductionmentioning

confidence: 99%

“…However, there is limited progress in utilizing the quasi-uniform random vectors for solving problems in coding and communications. This limited progress is mainly due to the observations that generating quasi-uniform random vectors is noted as a hard combinatorial problem [9], and an optimization problem over quasi-uniform random vectors is known to be extremely hard to solve [6]. In recent work, a recursive algorithm was given to verify whether a vector is the entropy vector of some quasi-uniform random vector [10].…”

Section: Introductionmentioning

confidence: 99%

A Quasi-Uniform Approach to Characterizing the Boundary of the Almost Entropic Region

Thakor¹,

Saleem²

2022

Preprint

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The convex closure of entropy vectors for quasiuniform random vectors is the same as the closure of the entropy region. Thus, quasi-uniform random vectors constitute an important class of random vectors for characterizing the entropy region. Moreover, the one-to-one correspondence between quasiuniform codes and quasi-uniform random vectors makes quasiuniform random vectors of central importance for designing effective codes for communication systems. In this paper, we present a novel approach that utilizes quasi-uniform random vectors for characterizing the boundary of the almost entropic region. In particular, we use the notion of quasi-uniform random vectors to establish looseness of known inner bounds for the entropy vectors at the boundary of the almost entropic region for three random variables. For communication models such as network coding, our approach can be applied to design network codes from quasi-uniform entropy vectors.

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MCMC methods for entropy optimization and nonlinear network coding

Cited by 12 publications

References 7 publications

On Ingleton-Violating Finite Groups

On Ingleton-Violating Finite Groups

Survey of Network Coding and Its Applications

A Quasi-Uniform Approach to Characterizing the Boundary of the Almost Entropic Region

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