On the Capacity of the Finite Field Counterparts of Wireless Interference Networks

This work explores how degrees of freedom (DoF) results from wireless networks can be translated into capacity results for their finite field counterparts that arise in network coding applications. The main insight is that scalar (SISO) finite field channels over $\mathbb{F}_{p^n}$ are analogous to n x n vector (MIMO) channels in the wireless setting, but with an important distinction -- there is additional structure due to finite field arithmetic which enforces commutativity of matrix multiplication and limits the channel diversity to n, making these channels similar to diagonal channels in the wireless setting. Within the limits imposed by the channel structure, the DoF optimal precoding solutions for wireless networks can be translated into capacity optimal solutions for their finite field counterparts. This is shown through the study of the 2-user X channel and the 3-user interference channel. Besides bringing the insights from wireless networks into network coding applications, the study of finite field networks over $\mathbb{F}_{p^n}$ also touches upon important open problems in wireless networks (finite SNR, finite diversity scenarios) through interesting parallels between p and SNR, and n and diversity.Comment: Full version of paper accepted for presentation at ISIT 201

Constraints on the thermal evolution of Venus inferred from Magellan data

The impact craters with diameters from 1.5 to 280 km compiled from Magellan observations indicate that the crater population on Venus has a completely spatially random distribution and the size/density distribution of craters with diameters greater than or equal to 35 km is consistent with a 'production' population with an age of 500 plus or minus 250 m.y. The similarity in size distribution from area to area indicates that the crater distribution is independent of crater size. Also, the forms of the modified craters are virtually identical to those of the pristine craters. These observations imply that Venus reset its cratering record by global resurfacing 500 m.y. ago, and resurfacing declined relatively fast. The fact that less than 40 percent of all craters have been modified and that the few volcanically embayed craters are located on localized tectonic regions indicate that only minor and localized volcanism and tectonism have occurred since the latest vigorous resurfacing event approximately 500 m.y. ago and the interior of Venus has been solid and possibly colder than Earth's. This is because the high-temperature lithosphere of Venus would facilitate upward ascending of mantle plumes and result in extensive volcanism if the venusian upper mantle were as hot as or hotter than Earth's. Therefore, the present surface morphology of Venus may provide useful constraints on the pattern of that vigorous convection, and possibly on the thermal state of the venusian mantle. We examine this possibility through numerical calculations of three-dimensional thermal convection models in a spherical shell with temperature- and pressure-dependent Newtonian viscosity, temperature-dependent thermal diffusivity, pressure-dependent thermal expansion coefficient, and time-dependent internal heat production rate solar magnitude

Distributed Data Storage with Minimum Storage Regenerating Codes - Exact and Functional Repair are Asymptotically Equally Efficient

We consider a set up where a file of size M is stored in n distributed storage nodes, using an (n,k) minimum storage regenerating (MSR) code, i.e., a maximum distance separable (MDS) code that also allows efficient exact-repair of any failed node. The problem of interest in this paper is to minimize the repair bandwidth B for exact regeneration of a single failed node, i.e., the minimum data to be downloaded by a new node to replace the failed node by its exact replica. Previous work has shown that a bandwidth of B=[M(n-1)]/[k(n-k)] is necessary and sufficient for functional (not exact) regeneration. It has also been shown that if k < = max(n/2, 3), then there is no extra cost of exact regeneration over functional regeneration. The practically relevant setting of low-redundancy, i.e., k/n>1/2 remains open for k>3 and it has been shown that there is an extra bandwidth cost for exact repair over functional repair in this case. In this work, we adopt into the distributed storage context an asymptotically optimal interference alignment scheme previously proposed by Cadambe and Jafar for large wireless interference networks. With this scheme we solve the problem of repair bandwidth minimization for (n,k) exact-MSR codes for all (n,k) values including the previously open case of k > \max(n/2,3). Our main result is that, for any (n,k), and sufficiently large file sizes, there is no extra cost of exact regeneration over functional regeneration in terms of the repair bandwidth per bit of regenerated data. More precisely, we show that in the limit as M approaches infinity, the ratio B/M = (n-1)/(k(n-k))$