Changho Suh

A Survey on Network Codes for Distributed Storage

Distributed storage systems often introduce redundancy to increase reliability. When coding is used, the repair problem arises: if a node storing encoded information fails, in order to maintain the same level of reliability we need to create encoded information at a new node. This amounts to a partial recovery of the code, whereas conventional erasure coding focuses on the complete recovery of the information from a subset of encoded packets. The consideration of the repair network traffic gives rise to new design challenges. Recently, network coding techniques have been instrumental in addressing these challenges, establishing that maintenance bandwidth can be reduced by orders of magnitude compared to standard erasure codes. This paper provides an overview of the research results on this topic.

Interference Alignment for Cellular Networks

In this paper, we propose a new way of interference management for cellular networks. We develop the scheme that approaches to interference-free degree-of-freedom (dof) as the number K of users in each cell increases. Also we find the corresponding bandwidth scaling conditions for typical wireless channels: multi-path channels and single-path channels with propagation delay. The scheme is based on interference alignment. Especially for more-than-two-cell cases where there are multiple non-intended BSs, we propose a new version of interference alignment, namely subspace interference alignment. The idea is to align interferences into multi-dimensional subspace (instead of one dimension) for simultaneous alignments at multiple non-intended BSs. The proposed scheme requires finite dimensions growing linearly with K, i.e., ~O(K).

Downlink Interference Alignment

We develop an interference alignment (IA) technique for a downlink cellular system. In the uplink, IA schemes need channel-state-information exchange across base-stations of different cells, but our downlink IA technique requires feedback only within a cell. As a result, the proposed scheme can be implemented with minimal changes to an existing cellular system where the feedback mechanism (within a cell) is already being considered for supporting multi-user MIMO. Not only is our proposed scheme implementable with little effort, it can in fact provide substantial gain especially when interference from a dominant interferer is significantly stronger than the remaining interference: it is shown that in the two-isolated cell layout, our scheme provides four-fold gain in throughput performance over a standard multi-user MIMO technique. We show through simulations that our technique provides respectable gain under a more realistic scenario: it gives approximately 20% gain for a 19 hexagonal wrap-around-cell layout.

Feedback Capacity of the Gaussian Interference Channel to Within 2 Bits

We characterize the capacity region to within 2 bits/s/Hz and the symmetric capacity to within 1 bit/s/Hz for the two-user Gaussian interference channel (IC) with feedback. We develop achievable schemes and derive a new outer bound to arrive at this conclusion. One consequence of the result is that feedback provides multiplicative gain at high signal-to-noise ratio: the gain becomes arbitrarily large for certain channel parameters. This finding is in contrast to point-to-point and multiple-access channels where feedback provides no gain and only bounded additive gain respectively. The result makes use of a linear deterministic model to provide insights into the Gaussian channel. This deterministic model is a special case of the El Gamal-Costa deterministic model and as a side-generalization, we establish the exact feedback capacity region of this general class of deterministic ICs.

Resource allocation for multicast services in multicarrier wireless communications

We consider a multicast resource allocation problem for the downlink in OFDM-based wireless cellular network systems. In a conventional multicast system, to accommodate users with bad channel conditions, the transmission is based on the worst case user. We show that such a multicast system saturates the capacity when the number of users increases in fading environments. We exploit the multicarrier nature of OFDM and advances in coding techniques such as MDC (multiple description coding), in which arbitrary combinations of layers can be decoded at the receiver. Different MDC layers are carried over different subcarriers and users with good channels receive data from more subcarriers than users with poor channel conditions. We present an optimal subcarrier/bit allocation method requiring full search of possible candidates. To reduce the complexity, we propose a two-step suboptimum algorithm by separating subcarrier allocation and bit loading. Numerical results show that the proposed heuristics significantly outperform the conventional multicast transmission scheme. The difference between optimum and heuristic solutions is less than 5%.

Exact-Repair MDS Code Construction Using Interference Alignment

The high repair cost of (n, k) Maximum Distance Separable (MDS) erasure codes has recently motivated a new class of MDS codes, called Repair MDS codes, that can significantly reduce repair bandwidth over conventional MDS codes. In this paper, we describe (n, k, d) Exact-Repair MDS codes, which allow for any failed node to be repaired exactly with access to d survivor nodes, where k ≤ d ≤ n-1. We construct Exact-Repair MDS codes that are optimal in repair bandwidth for the cases of: (α) k/n ≤ 1/2 and d ≥ 2k - 11; (b) k ≤ 3. Our codes are deterministic and require a finite-field size of at most 2(n - k). Our constructive codes are based on interference alignment techniques.

Asymptotic Interference Alignment for Optimal Repair of MDS Codes in Distributed Storage

The high repair bandwidth cost of (<i>n</i>,<i>k</i>) maximum distance separable (MDS) erasure codes has motivated a new class of codes that can reduce repair bandwidth over that of conventional MDS codes. In this paper, we address (<i>n</i>,<i>k</i>,<i>d</i>) exact repair MDS codes, which allow for any single failed node to be repaired exactly with access to any arbitrary set of <i>d</i> survivor nodes. We show the existence of exact repair MDS codes that achieve minimum repair bandwidth (matching the cut-set lower bound) for arbitrary admissible (<i>n</i>,<i>k</i>,<i>d</i>), i.e., <i>k</i> ≤ <i>d</i> ≤ <i>n</i>-1. Moreover, we extend our results to show the optimality of our codes for multiple-node failure scenarios in which an arbitrary set of <i>r</i> ≤ <i>n</i>-<i>k</i> failed nodes needs to repaired. Our approach is based on asymptotic interference alignment proposed by Cadambe and Jafar. As a byproduct, we also characterize the capacity of a class of multisource nonmulticast networks.

Exact-repair MDS codes for distributed storage using interference alignment

The high repair cost of (n, k) Maximum Distance Separable (MDS) erasure codes has recently motivated a new class of codes, called Regenerating Codes, that optimally trade off storage cost for repair bandwidth. In this paper, we address bandwidth-optimal (n, k, d) Exact-Repair MDS codes, which allow for any failed node to be repaired exactly with access to arbitrary d survivor nodes, where k ≤ d ≤ n − 1. Under scalar-linear codes which do not permit symbol-splitting, we construct Exact-Repair MDS codes that are optimal in repair bandwidth for the case of k/n ≤ 1/2 and d ≥ 2k − 1. Our codes are deterministic and require a finite-field size of at most 2(n − k). Under vector-linear codes which allow for the break-up of stored symbols into arbitrarily small subsymbols, we show the existence of optimal Exact-Repair codes for the entire admissible range of possible (n, k, d), i.e., k < n and k ≤ d ≤ n − 1. That is, we establish the existence of vector-linear Exact-Repair MDS codes that match the fundamental cutset lower bound. Our approach for both the constructive scalar-linear code design and for the existence of vector-linear codes is based on interference alignment techniques.

Resource Allocation for Multicast Services in Multicarrier Wireless Communications

We consider a multicast resource allocation problem for the downlink in OFDM-based wireless cellular network systems. In a conventional multicast system, to accommodate users with bad channel conditions, the transmission is based on the worst case user. We show that such a multicast system saturates the capacity when the number of users increases in fading environments. We exploit the multicarrier nature of OFDM and advances in coding techniques such as MDC (multiple description coding), in which arbitrary combinations of layers can be decoded at the receiver. Different MDC layers are carried over different subcarriers and users with good channels receive data from more subcarriers than users with poor channel conditions. We present an optimal subcarrier/bit allocation method requiring full search of possible candidates. To reduce the complexity, we propose a two-step suboptimum algorithm by separating subcarrier allocation and bit loading. Numerical results show that the proposed heuristics significantly outperform the conventional multicast transmission scheme. The difference between optimum and heuristic solutions is less than 5%.

Linear Degrees of Freedom of the MIMO X-Channel With Delayed CSIT

We establish the degrees of freedom (DoF) of the two-user X-channel with delayed channel knowledge at transmitters [i.e., delayed channel state information at the transmitters (CSIT)], assuming linear coding strategies at the transmitters. We derive a new upper bound and characterize the linear DoF of this network to be 6/5. The converse builds upon our development of a general lemma that shows that, if two distributed transmitters employ linear strategies, the ratio of the dimensions of received linear subspaces at the two receivers cannot exceed 3/2, due to delayed CSIT. As a byproduct, we also apply this general lemma to the three-user interference channel with delayed CSIT, thereby deriving a new upper bound of 9/7 on its linear DoF. This is the first bound that captures the impact of delayed CSIT on the DoF of this network, under the assumption of linear encoding strategies.