When there are multiple channels between a pair of source and destination, the way to use the channels carries some extra information. In this paper, we study a coding scheme that encodes the message into the choices of channels. Unlike the traditional sense that the message bits are ordered in a temporal manner, the ordering information is lost after the transformation. To resolve this issue, the data to be sent via the channels is the ordering of the message blocks. We study the properties on the number of channel-uses and the limitation of the code. We also investigate the optimal message length that maximizes the efficiency of channel-uses.

Zero-error single-channel source coding has been studied extensively over the past decades. Its natural multi-channel generalization is however seldom investigated. While the special case with multiple symmetric-alphabet channels was studied a decade ago, codes in such setting have no advantage over single-channel codes in data compression, making them worthless in most applications. With essentially no development since the last decade, in this paper, we break the stalemate by showing that it is possible to beat single-channel source codes in terms of compression assuming asymmetric-alphabet channels. We present the multi-channel analogs of several classical results in single-channel source coding, e.g., a multi-channel Huffman code is an optimal tree-decodable code. We also show evidences that finding an efficient construction of multi-channel Huffman codes may be hard. Nevertheless, we propose a construction whose redundancy is guaranteed to be no larger than that of an optimal single-channel source code.

Although n-channel prefix-free codes are natural extensions of their 1-channel counterpart, the extra dimensions greatly increase the complexity of the problem such that most classical results cannot be generalized directly. Recently, a greedy algorithm was developed for deciding if it is possible to construct a 2-channel prefix-free code from a given multiset of codeword lengths. By dropping the information about codeword assignments which are not necessary for the decision problem, the greedy algorithm runs in polynomial time. However, if we naively turn the decision algorithm into a search algorithm (for constructing a prefix-free code) by retaining the information about codeword assignments, the computational complexity becomes exponential. One puzzle left unsolved was that whether the search problem can also be solved in polynomial time. In this paper, we give an affirmative answer to this question by designing a tailor-made data structure and a new lazy evaluation technique.

Batched network coding is a practical branch of random linear network coding which encodes the input file into small batches of coded packets. Although burst erasures within the batches can reduce the advantage of network coding, it can be alleviated by applying a block interleaver which spreads the burst across multiple batches. On the other hand, the intermediate network nodes are required to perform recoding on the batches unlike the traditional forwarding strategy. The recoding of a batch can be started once all the packets in that batch which are not dropped by the channel are received. This means that the intermediate network nodes have to deinterleave the batches for recoding and reinterleave the batches again for transmission, which gives us a choice to use different interleaver depths at different network nodes. In this paper, we study the memory requirements of the schemes for transmitting batches which apply different interleaver depths. We investigate the periodic structure of buffer sizes and the connection between buffer sizes and the total delay induced by the interleaver. More importantly, we show that there exists a scheme which can achieve the lowest memory requirement and minimum total delay simultaneously.

We study a general Omniscient Arbitrarily Varying Channel (AVC) problem where Alice wishes to communicate a message to receiver Bob by inputting a length-n vector x to a channel. Jammer James observes x, and as a function of x chooses a state sequence s. Bob observes y (such that channel inputs and outputs are related component-wise as yi= w(xi,si) for some deterministic function w(.,.)) from which he must estimate m with no error. Input and state constraints determine feasible inputs x and s for Alice and James respectively. In this work we characterize when a positive communication rate is possible.We first show that the capacity of any such AVC completely depends upon the relationship between a confusability set, and the set of completely-positive-self-couplings (both are convex sets of certain single-letter probability distributions). Our main result provides essentially matching necessary and sufficient conditions for capacity positivity; we show that the zero-error capacity of an AVC is positive if there are completely-positive-self-couplings outside the confusability set of the given AVC; and that the AVC capacity is zero if all completely-positive-self couplings are in the interior of this confusability set. Our achievability uses a novel code construction based on completely-positive-self-couplings called cloud codes which are strict generalizations of all known Gilbert-Varshamov (GV) type codes. Our converse is based upon Ramsey-theoretic ideas, a generalization of the Plotkin bound leveraging a known result on the duality of completely positive matrices and copositive matrices, and a Fourier-analytic proof of the non-existence of certain sequences of random variables.

The Kraft inequality gives a necessary and sufficient condition for the existence of a single channel prefix-free code. However, the multichannel Kraft inequality does not imply the existence of a multichannel prefix-free code in general. It is natural to ask whatever there exists an efficient decision procedure for the existence of multichannel prefix-free codes. In this paper, we tackle the two-channel case of the above problem by relating it to a constrained rectangle packing problem. Although a general rectangle packing problem is NP-complete, the extra imposed constraints allow us to propose an algorithm which can solve the problem efficiently.

Batched network coding is a practical realization of random linear network coding, which encodes the packets for transmission into small batches of coded packets. The advantage of network coding can be reduced when a burst loss occurs. Block interleaver is one of the approach to spread the burst across multiple batches. However, an intermediate node has to receive all the packets in a batch which are not dropped by the channel before it can perform recoding on the batch, which means that the node has to deinterleave for recoding and reinterleave again for transmission. This gives us the freedom to select heterogeneous interleaver depth among the intermediate nodes. In general, the larger the interleaver depth, the better the spread of the bursts. On the other hand, we want the delay as short as possible to enable real-time applications. In this paper, we investigate the delay induced by applying a block interleaver on batched network codes. We also show that a homogeneous interleaver depth is the largest interleaver depth we can use to achieve a minimum delay.

Batched network coding is a variation of random linear network coding which has low computational and storage costs. In order to adapt random fluctuations in the number of erasures in individual batches, it is not optimal to recode and transmit the same number of packets for all batches. Different distributed optimization problems, which are called adaptive recoding, were formulated for this purpose. The key component of these optimization problems is the expected value of the rank distribution of a batch at the next network node, which also known as the expected rank. In this paper, we put forth a unified adaptive recoding framework. We show that the expected rank functions are concave when the packet loss pattern follows a stationary stochastic process regardless of the field size, which covers but not limited to independent packet loss and burst packet loss. Under this concavity property, we show that there always exists a preferred solution which not only can make the number of recoded packets almost deterministic but can also tolerate rank distribution errors due to inaccurate measurements or limited precision of the machine. To obtain such an optimal solution, we propose tuning schemes that can turn any feasible solution into one with the above desired properties.