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

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))$

Multiple Access Outerbounds and the Inseparability of Parallel Interference Channels

It is known that the capacity of parallel (multi-carrier) Gaussian point-to-point, multiple access and broadcast channels can be achieved by separate encoding for each subchannel (carrier) subject to a power allocation across carriers. In this paper we show that such a separation does not apply to parallel Gaussian interference channels in general. A counter-example is provided in the form of a 3 user interference channel where separate encoding can only achieve a sum capacity of $\log({SNR})+o(\log({SNR}))$ per carrier while the actual capacity, achieved only by joint-encoding across carriers, is $3/2\log({SNR}))+o(\log({SNR}))$ per carrier. As a byproduct of our analysis, we propose a class of multiple-access-outerbounds on the capacity of the 3 user interference channel

Degrees of Freedom of Wireless X Networks

We explore the degrees of freedom of $M\times N$ user wireless $X$ networks, i.e. networks of $M$ transmitters and $N$ receivers where every transmitter has an independent message for every receiver. We derive a general outerbound on the degrees of freedom \emph{region} of these networks. When all nodes have a single antenna and all channel coefficients vary in time or frequency, we show that the \emph{total} number of degrees of freedom of the $X$ network is equal to $\frac{MN}{M+N-1}$ per orthogonal time and frequency dimension. Achievability is proved by constructing interference alignment schemes for $X$ networks that can come arbitrarily close to the outerbound on degrees of freedom. For the case where either M=2 or N=2 we find that the outerbound is exactly achievable. While $X$ networks have significant degrees of freedom benefits over interference networks when the number of users is small, our results show that as the number of users increases, this advantage disappears. Thus, for large $K$, the $K\times K$ user wireless $X$ network loses half the degrees of freedom relative to the $K\times K$ MIMO outerbound achievable through full cooperation. Interestingly, when there are few transmitters sending to many receivers ($N\gg M$) or many transmitters sending to few receivers ($M\gg N$), $X$ networks are able to approach the $\min(M,N)$ degrees of freedom possible with full cooperation on the $M\times N$ MIMO channel. Similar to the interference channel, we also construct an example of a 2 user $X$ channel with propagation delays where the outerbound on degrees of freedom is achieved through interference alignment based on a simple TDMA strategy.Comment: 26 page