We consider the locally repairable codes (LRCs), aiming at sequentially recovering multiple erasures; in particular, we propose and study the so-called (n, k, r, t)-sequential LRCs (SLRC) as an [n, k] linear code, where any t' (≤ t) erasures can be sequentially recovered, each by r (2 ≤ r <; k) other code symbols. Here, sequential recovering means that the erased symbols are recovered one by one, and an already recovered symbol can be used to recover the remaining erased symbols. This important recovering method, in contrast with the extensively studied parallel recovering, is currently far from being thoroughly understood; more specifically, there are to date no codes constructed for arbitrary t ≥ 3 erasures and bounds to evaluate the performance of such codes. We first derive a tight upper bound on the code rate of the (n, k, r, t)-SLRC for t = 3 and r ≥ 2. We then propose two constructions of binary (n, k, r, t)-SLRCs for general r, t ≥ 2 (existing constructions only deal with t ≤7 erasures). The first construction generalizes the method of direct product construction. The second construction is based on the resolvable configurations and yields SLRCs for any r ≥ 2 odd t ≥ 3. For both constructions, the rates are optimal for t ∈ {2, 3} and are higher than most of the existing LRC families for arbitrary t ≥ 4.

Unveiling a fundamental link between information theory and estimation theory, the I-MMSE relationship by Guo et al., together with its numerous extensions, has great theoretical significance and various practical applications. On the other hand, its influences to date have been restricted to channels without feedback or memory, due to the absence of its extensions to such channels. In this paper, we propose the extensions of the I-MMSE relationship to discrete-time and continuous-time Gaussian channels with feedback and/or memory. Our approach is based on a very simple observation, which can be applied to other scenarios, such as a simple and direct proof of the classical de Bruijn's identity.

We consider a memoryless channel with an input Markov process supported on a mixing finite-type constraint. We continue the development of asymptotics for the entropy rate of the output hidden Markov chain and deduce that, at high signal-to-noise ratio, the mutual information rate of such a channel is concave with respect to “almost” all input Markov chains of a given order.

In this paper, we examine an input-constrained erasure channel and we characterize the asymptotics of its capacity when the erasure rate is low. More specifically, for a general memoryless erasure channel with its input supported on an irreducible finite-type constraint, we derive partial asymptotics of its capacity, using some series expansion type formula of its mutual information rate; and for a binary erasure channel with its first-order Markovian input supported on the $(1, \infty )$ -RLL constraint based on the concavity of its mutual information rate with respect to some parameterization of the input, we numerically evaluate its first-order Markov capacity and further derive its full asymptotics. The asymptotics obtained in this paper, when compared with the recently derived feedback capacity for a binary erasure channel with the same input constraint, enable us to draw the conclusion that feedback may increase the capacity of an input-constrained channel, even if the channel is memoryless.

It is well known that the problem of computing the feedback capacity of a stationary Gaussian channel can be recast as an infinite-dimensional optimization problem; moreover, necessary and sufficient conditions for the optimality of a solution to this optimization problem have been characterized, and based on this characterization, an explicit formula for the feedback capacity has been given for the case that the noise is a first-order autoregressive moving-average Gaussian process. In this paper, via a simple “change of variables” trick, we further examine the above-mentioned infinite-dimensional optimization problem. We prove that unless the Gaussian noise is white, its optimal solution is unique, and we propose an algorithm to recursively compute the unique optimal solution, which is guaranteed to converge in theory and features an efficient implementation for a suboptimal solution in practice. Furthermore, for the case that the noise is a k-th order autoregressive moving-average Gaussian process, we give a relatively more explicit formula for the feedback capacity; more specifically, the feedback capacity is expressed as a simple function evaluated at a solution to a system of polynomial equations.

In this paper, we propose to use Brownian motions to formulate continuous-time multiuser Gaussian networks and derive the capacity regions of a continuous-time white Gaussian multiple access channel with/without feedback, a continuous-time white Gaussian interference channel without feedback and a continuous-time white Gaussian broadcast channel without feedback. These “complete” results stand in stark contrast to the status quo of network information theory in discrete-time, where the capacity regions of the all the above-mentioned channels are known only for a handful of special scenarios. For certain cases, our results echo, from a different perspective, the folklore that “a continuous-time channel is the limit of bandwidth limited discrete-time ones as the bandwidth tends to infinity”.