Let $mathbf{H}$ be the cartesian product of a family of finite abelian groups indexed by a finite set $Omega$. A given poset (i.e., partially ordered set) $mathbf{P}=(Omega,preccurlyeq_{mathbf{P}})$ gives rise to a poset metric on $mathbf{H}$, which
further leads to a partition $mathcal{Q}(mathbf{H},mathbf{P})$ of $mathbf{H}$. We prove that if $mathcal{Q}(mathbf{H},mathbf{P})$ is Fourier-reflexive, then its dual partition $Lambda$ coincides with the partition of $hat{mathbf{H}}$ induced by $mathbf{overline{P}}$, the dual poset of $mathbf{P}$, and moreover, $mathbf{P}$ is necessarily hierarchical. This result establishes a conjecture proposed by Gluesing-Luerssen in cite{4}. We also show that with some other assumptions, $Lambda$ is finer than the partition of $hat{mathbf{H}}$ induced by $mathbf{overline{P}}$. In addition, we give some necessary and sufficient conditions for $mathbf{P}$ to be hierarchical, and for the case that $mathbf{P}$ is hierarchical, we give an explicit criterion for determining whether two codewords in $hat{mathbf{H}}$ belong to the same block of $Lambda$. We prove these results by relating the involved partitions with certain family of polynomials, a generalized version of which is also proposed and studied to generalize the aforementioned results.
The Langberg-M{e}dard multiple unicast conjecture claims that for a strongly reachable $k$-pair network, there exists a multi-flow with rate $(1,1,dots,1)$. In this paper, we show that the conjecture holds true for {em stable} $3$-pair networks.
We propose two modifi
A renowned information-theoretic formula by Shannon expresses the mutual information rate of a white Gaussian channel with a stationary Gaussian input as an integral of a simple function of the power spectral density of the channel input. We give in
this paper a rigorous yet elementary proof of this classical formula. As opposed to all the conventional approaches, which either rely on heavy mathematical machineries or have to resort to some external results, our proof, which hinges on a recently proven sampling theorem, is elementary and self-contained, only using some well-known facts from basic calculus and matrix theory.
The Langberg-Medard multiple unicast conjecture claims that for any strongly reachable $k$-pair network, there exists a multi-flow with rate $(1,1,dots,1)$. In a previous work, through combining and concatenating the so-called elementary flows, we ha
ve constructed a multi-flow with rate at least $(frac{8}{9}, frac{8}{9}, dots, frac{8}{9})$ for any $k$. In this paper, we examine an optimization problem arising from this construction framework. We first show that our previous construction yields a sequence of asymptotically optimal solutions to the aforementioned optimization problem. And furthermore, based on this solution sequence, we propose a perturbation framework, which not only promises a better solution for any $k mod 4 eq 2$ but also solves the optimization problem for the cases $k=3, 4, dots, 10$, accordingly yielding multi-flows with the largest rate to date.
