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Network coding enhances performance in network communications and distributed storage by increasing throughput and robustness while reducing latency. Batched Sparse (BATS) codes are a class of capacity-achieving network codes, but their practical applications are hindered by their structure, computational intensity, and power demands of finite field operations. Most literature focuses on algorithmic-level techniques to improve coding efficiency. Optimization with an algorithm/hardware co-designing approach has long been neglected. Leveraging the unique structure of BATS codes, we first present CS-BATS, a hardware-friendly variant. Next we propose a simple but effective bounded-value generator, to reduce the size of a finite field multiplier by up to 70%. Finally, we report on a scalable and resource-efficient FPGA-based network coding accelerator that achieves a throughput of 27 Gbps, a speedup of more than 300 over software.

It is with heavy hearts that we mourn the passing of Ning Cai, a luminary whose pioneering spirit illuminated the realms of network coding and beyond. On May 25, 2023, at the age of 75, Prof. Cai bid farewell, leaving behind a profound legacy that continues to resonate across generations of researchers. His contributions spanned a vast spectrum, from the groundbreaking explorations in network coding to the intricate realms of quantum information theory. Ning's indelible mark on the academic landscape is a testament to his unwavering dedication and relentless pursuit of knowledge. Among his many accolades, Ning's seminal works garnered widespread recognition, exemplified by the prestigious 2005 IEEE Information Theory Society Paper Award for his work "Linear Network Coding." Furthermore, his enduring impact was underscored by the 2018 ACM SIGMOBILE Test-of-Time Paper Award, bestowed upon his paper "Network Information Flow." In addition to his scholarly achievements, Ning's unwavering commitment to mentorship has left an indelible mark on countless aspiring scholars. His guidance and wisdom continue to inspire and guide future generations in their scholarly pursuits. As we bid farewell to a titan in the field, let us cherish the legacy of Ning Cai, whose brilliance and generosity of spirit will forever endure in the annals of academia.

In coding theory, codes are usually designed with a certain level of randomness to facilitate analysis and accommodate different channel conditions. However, the resulting random code constructed can be suboptimal in practical implementations. Represented by a bipartite graph, the Batched Sparse Code (BATS Code) is a randomly constructed erasure code that utilizes network coding to achieve near-optimal performance in wireless multi-hop networks. In the performance analysis in the previous research, it is implicitly assumed that the coded batches in the BATS code are independent. This assumption holds only asymptotically when the number of input symbols is infinite, but it does not generally hold in a practical setting where the number of input symbols is finite, especially when the code is constructed randomly. We show that dependence among the batches significantly degrades the code's performance. In order to control the batch dependence through graphical design, we propose constructing the BATS code in a structured manner. A hardware-friendly structured BATS code called the Cyclic-Shift BATS (CS-BATS) code is proposed, which constructs the code from a small base graph using light-weight cyclic-shift operations. We demonstrate that when the base graph is properly designed, a higher decoding rate and a smaller complexity can be achieved compared with the random BATS code.

The proof of information inequalities and identities under linear constraints on the information measures is an important problem in information theory. For this purpose, ITIP and other variant algorithms have been developed and implemented, which are all based on solving a linear program (LP). In this paper, we develop a method with symbolic computation. Compared with the known methods, our approach can completely avoids the use of linear programming which may cause numerical errors. Our procedures are also more efficient computationally., Comment: arXiv admin note: text overlap with arXiv:2202.02786

We consider a generalized collision channel model for general multi-user communication systems, an extension of Massey and Mathys' collision channel without feedback for multiple access communications. In our model, there are multiple transmitters and receivers sharing the same communication channel. The transmitters are not synchronized and arbitrary time offsets between transmitters and receivers are assumed. A ``collision" occurs if two or more packets from different transmitters partially or completely overlap at a receiver. Our model includes the original collision channel as a special case. This paper focuses on reliable throughputs that are approachable for arbitrary time offsets. We consider both slot-synchronized and non-synchronized cases and characterize their reliable throughput regions for the generalized collision channel model. These two regions are proven to coincide. Moreover, it is shown that the protocol sequences constructed for multiple access communication remain ``throughput optimal" in the generalized collision channel model. We also identify the protocol sequences that can approach the outer boundary of the reliable throughput region.

In this paper, we put forward secure network function computation over a directed acyclic network. In such a network, a sink node is required to compute with zero error a target function of which the inputs are generated as source messages at multiple source nodes, while a wiretapper, who can access any one but not more than one wiretap set in a given collection of wiretap sets, is not allowed to obtain any information about a security function of the source messages. The secure computing capacity for the above model is defined as the maximum average number of times that the target function can be securely computed with zero error at the sink node with the given collection of wiretap sets and security function for one use of the network. The characterization of this capacity is in general overwhelmingly difficult. In the current paper, we consider securely computing linear functions with a wiretapper who can eavesdrop any subset of edges up to a certain size r, referred to as the security level, with the security function being the identity function. We first prove an upper bound on the secure computing capacity, which is applicable to arbitrary network topologies and arbitrary security levels. When the security level r is equal to 0, our upper bound reduces to the computing capacity without security consideration. We discover the surprising fact that for some models, there is no penalty on the secure computing capacity compared with the computing capacity without security consideration. We further obtain an equivalent expression of the upper bound by using a graph-theoretic approach, and accordingly we develop an efficient approach for computing this bound. Furthermore, we present a construction of linear function-computing secure network codes and obtain a lower bound on the secure computing capacity.

Proving linear inequalities and identities of Shannon's information measures, possibly with linear constraints on the information measures, is an important problem in information theory. For this purpose, ITIP and other variant algorithms have been developed and implemented, which are all based on solving a linear program (LP). In particular, an identity $f = 0$ is verified by solving two LPs, one for $f \ge 0$ and one for $f \le 0$. In this paper, we develop a set of algorithms that can be implemented by symbolic computation. Based on these algorithms, procedures for verifying linear information inequalities and identities are devised. Compared with LP-based algorithms, our procedures can produce analytical proofs that are both human-verifiable and free of numerical errors. Our procedures are also more efficient computationally. For constrained inequalities, by taking advantage of the algebraic structure of the problem, the size of the LP that needs to be solved can be significantly reduced. For identities, instead of solving two LPs, the identity can be verified directly with very little computation.

Batched network coding (BNC) is a low-complexity solution to network transmission in multi-hop packet networks with packet loss. BNC encodes the source data into batches of packets. As a network coding scheme, the intermediate nodes perform recoding on the received packets belonging to the same batch instead of just forwarding them. A recoding scheme that may generate more recoded packets for batches of a higher rank is also called adaptive recoding. Meanwhile, in order to combat burst packet loss, the transmission of a block of batches can be interleaved. Stream interleaving studied in literature achieves the maximum separation among any two consecutive packets of a batch, but permutes packets across blocks and hence cannot bound the buffer size and the latency. To resolve the issue of stream interleaver, we design an intrablock interleaver for adaptive recoding that can preserve the advantages of using a block interleaver when the number of recoded packets is the same for all batches. We use potential energy in classical mechanics to measure the performance of an interleaver, and propose an algorithm to optimize the interleaver with this performance measure. Our problem formulation and algorithm for intrablock interleaving are also of independent interest., Comment: This paper was presented in part at 2021 IEEE International Symposium on Information Theory

We consider linear network error correction (LNEC) coding when errors may occur on edges of a communication network of which the topology is known. In this paper, we first revisit and explore the framework of LNEC coding, and then unify two well-known LNEC coding approaches. Furthermore, by developing a graph-theoretic approach to the framework of LNEC coding, we obtain a significantly enhanced characterization of the error correction capability of LNEC codes in terms of the minimum distances at the sink nodes. In LNEC coding, the minimum required field size for the existence of LNEC codes, in particular LNEC maximum distance separable (MDS) codes which are a type of most important optimal codes, is an open problem not only of theoretical interest but also of practical importance, because it is closely related to the implementation of the coding scheme in terms of computational complexity and storage requirement. By applying the graph-theoretic approach, we obtain an improved upper bound on the minimum required field size. The improvement over the existing results is in general significant. The improved upper bound, which is graph-theoretic, depends only on the network topology and requirement of the error correction capability but not on a specific code construction. However, this bound is not given in an explicit form. We thus develop an efficient algorithm that can compute the bound in linear time. In developing the upper bound and the efficient algorithm for computing this bound, various graph-theoretic concepts are introduced. These concepts appear to be of fundamental interest in graph theory and they may have further applications in graph theory and beyond., Comment: 41 pages

Multilevel diversity coding is a classical coding model where multiple mutually independent information messages are encoded, such that different reliability requirements can be afforded to different messages. It is well known that {\em superposition coding}, namely separately encoding the independent messages, is optimal for symmetric multilevel diversity coding (SMDC) (Yeung-Zhang 1999). In the current paper, we consider weakly secure SMDC where security constraints are injected on each individual message, and provide a complete characterization of the conditions under which superposition coding is sum-rate optimal. Two joint coding strategies, which lead to rate savings compared to superposition coding, are proposed, where some coding components for one message can be used as the encryption key for another. By applying different variants of Han's inequality, we show that the lack of opportunity to apply these two coding strategies directly implies the optimality of superposition coding. It is further shown that under a set of particular security constraints, one of the proposed joint coding strategies can be used to construct a code that achieves the optimal rate region., Comment: The paper has been accepted by IEEE Transactions on Information Theory

Information-theoretic security is considered in the paradigm of network coding in the presence of wiretappers, who can access one arbitrary edge subset up to a certain size, also referred to as the security level. Secure network coding is applied to prevent the leakage of the source information to the wiretappers. In this two-part paper, we consider the problem of secure network coding when the information rate and the security level can change over time. In the current paper (i.e., Part I of the two-part paper), we focus on the problem for a fixed security level and a flexible rate. To efficiently solve this problem, we put forward local-encoding-preserving secure network coding, where a family of secure linear network codes (SLNCs) is called local-encoding-preserving if all the SLNCs in this family share a common local encoding kernel at each intermediate node in the network. We present an efficient approach for constructing upon an SLNC that exists a local-encoding-preserving SLNC with the same security level and the rate reduced by one. By applying this approach repeatedly, we can obtain a family of local-encoding-preserving SLNCs with a fixed security level and multiple rates. We also develop a polynomial-time algorithm for efficient implementation of this approach. Furthermore, it is proved that the proposed approach incurs no penalty on the required field size for the existence of SLNCs in terms of the best known lower bound by Guang and Yeung. The result in this paper will be used as a building block for efficiently constructing a family of local-encoding-preserving SLNCs for all possible pairs of rate and security level, which will be discussed in the companion paper (i.e., Part II of the two-part paper)., Comment: 32 pages

In the two-part paper, we consider the problem of secure network coding when the information rate and the security level can change over time. To efficiently solve this problem, we put forward local-encoding-preserving secure network coding, where a family of secure linear network codes (SLNCs) is called local-encoding-preserving (LEP) if all the SLNCs in this family share a common local encoding kernel at each intermediate node in the network. In this paper (Part II), we first consider the design of a family of LEP SLNCs for a fixed rate and a flexible security level. We present a novel and efficient approach for constructing upon an SLNC that exists an LEP SLNC with the same rate and the security level increased by one. Next, we consider the design of a family of LEP SLNCs for a fixed dimension (equal to the sum of rate and security level) and a flexible pair of rate and security level. We propose another novel approach for designing an SLNC such that the same SLNC can be applied for all the rate and security-level pairs with the fixed dimension. Also, two polynomial-time algorithms are developed for efficient implementations of our two approaches, respectively. Furthermore, we prove that both approaches do not incur any penalty on the required field size for the existence of SLNCs in terms of the best known lower bound by Guang and Yeung. Finally, we consider the ultimate problem of designing a family of LEP SLNCs that can be applied to all possible pairs of rate and security level. By combining the construction of a family of LEP SLNCs for a fixed security level and a flexible rate (obtained in Part I) with the constructions of the two families of LEP SLNCs in the current paper in suitable ways, we can obtain a family of LEP SLNCs that can be applied for all possible pairs of rate and security level. Three possible such constructions are presented., Comment: 38 pages

The problem of exact-repair regenerating codes against eavesdropping attack is studied. The eavesdropping model we consider is that the eavesdropper has the capability to observe the data involved in the repair of a subset of $\ell$ nodes. An $(n,k,d,\ell)$ secure exact-repair regenerating code is an $(n,k,d)$ exact-repair regenerating code that is secure under this eavesdropping model. It has been shown that for some parameters $(n,k,d,\ell)$, the associated optimal storage-bandwidth tradeoff curve, which has one corner point, can be determined. The focus of this paper is on characterizing such parameters. We establish a lower bound $\hat{\ell}$ on the number of wiretap nodes, and show that this bound is tight for the case $k = d = n-1$., Comment: This paper will be presented in part in ISIT 2018

It is well known that {\em superposition coding}, namely separately encoding the independent sources, is optimal for symmetric multilevel diversity coding (SMDC) (Yeung-Zhang 1999). However, the characterization of the coding rate region therein involves uncountably many linear inequalities and the constant term (i.e., the lower bound) in each inequality is given in terms of the solution of a linear optimization problem. Thus this implicit characterization of the coding rate region does not enable the determination of the achievability of a given rate tuple. In this paper, we first obtain closed-form expressions of these uncountably many inequalities. Then we identify a finite subset of inequalities that is sufficient for characterizing the coding rate region. This gives an explicit characterization of the coding rate region. We further show by the symmetry of the problem that only a much smaller subset of this finite set of inequalities needs to be verified in determining the achievability of a given rate tuple. Yet, the cardinality of this smaller set grows at least exponentially fast with $L$. We also present a subset entropy inequality, which together with our explicit characterization of the coding rate region, is sufficient for proving the optimality of superposition coding.

The problem of network function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node desires to compute with zero error a {\em target function}, of which the inputs are generated at multiple source nodes. The edges in the network are assumed to be error-free and have limited capacity. The nodes in the network are assumed to have unbounded computing capability and be able to perform network coding. The {\em computing rate} of a network code that can compute the target function over the network is the average number of times that the target function is computed with zero error for one use of the network. In this paper, we obtain an improved upper bound on the computing capacity, which is applicable to arbitrary target functions and arbitrary network topologies. This improved upper bound not only is an enhancement of the previous upper bounds but also is the first tight upper bound on the computing capacity for computing an arithmetic sum over a certain non-tree network, which has been widely studied in the literature. We also introduce a multi-dimensional array approach that facilitates evaluation of the improved upper bound. Furthermore, we apply this bound to the problem of computing a vector-linear function over a network. With this bound, we are able to not only enhance a previous result on computing a vector-linear function over a network but also simplify the proof significantly. Finally, we prove that for computing the binary maximum function over the reverse butterfly network, our improved upper bound is not achievable. This result establishes that in general our improved upper bound is non achievable, but whether it is asymptotically achievable or not remains open., Comment: 43 pages

We consider a communication network where there exist wiretappers who can access a subset of channels, called a wiretap set, which is chosen from a given collection of wiretap sets. The collection of wiretap sets can be arbitrary. Secure network coding is applied to prevent the source information from being leaked to the wiretappers. In secure network coding, the required alphabet size is an open problem not only of theoretical interest but also of practical importance, because it is closely related to the implementation of such coding schemes in terms of computational complexity and storage requirement. In this paper, we develop a systematic graph-theoretic approach for improving Cai and Yeung's lower bound on the required alphabet size for the existence of secure network codes. The new lower bound thus obtained, which depends only on the network topology and the collection of wiretap sets, can be significantly smaller than Cai and Yeung's lower bound. A polynomial-time algorithm is devised for efficient computation of the new lower bound., Comment: 35 pages

Let $X_i, i \in V$ form a Markov random field (MRF) represented by an undirected graph $G = (V,E)$, and $V'$ be a subset of $V$. We determine the smallest graph that can always represent the subfield $X_i, i \in V'$ as an MRF. Based on this result, we obtain a necessary and sufficient condition for a subfield of a Markov tree to be also a Markov tree. When $G$ is a path so that $X_i, i \in V$ form a Markov chain, it is known that the $I$-Measure is always nonnegative and the information diagram assumes a very special structure Kawabata and Yeung (1992). We prove that Markov chain is essentially the only MRF such that the $I$-Measure is always nonnegative. By applying our characterization of the smallest graph representation of a subfield of an MRF, we develop a recursive approach for constructing information diagrams for MRFs. Our work is built on the set-theoretic characterization of an MRF in Yeung, Lee, and Ye (2002).

Let $\cal{N}=\{1,\cdots,n\}$. The entropy function $\bf h$ of a set of $n$ discrete random variables $\{X_i:i\in\cal N\}$ is a $2^n$-dimensional vector whose entries are ${\bf{h}}({\cal{A}})\triangleq H(X_{\cal{A}}),\cal{A}\subset{\cal N} $, the (joint) entropies of the subsets of the set of $n$ random variables with $H(X_\emptyset)=0$ by convention. The set of all entropy functions for $n$ discrete random variables, denoted by $\Gamma^*_n$, is called the entropy function region for $n$. Characterization of $\Gamma^*_n$ and its closure $\overline{\Gamma^*_n}$ are well-known open problems in information theory. They are important not only because they play key roles in information theory problems but also they are related to other subjects in mathematics and physics. In this paper, we consider \emph{partition-symmetrical entropy functions}. Let $p=\{\cal{N}_1,\cdots, \cal{N}_t\}$ be a $t$-partition of $\cal N$. An entropy function $\bf h$ is called $p$-symmetrical if for all ${\cal A},{\cal B} \subset {\cal N}$, $\bf{h}({\cal A}) = \bf{h}({\cal B})$ whenever $|{\cal A} \cap {\cal N}_i| = |{\cal B} \cap {\cal N}_i|$, $i = 1, \cdots,t$. The set of all the $p$-symmetrical entropy functions, denoted by $\Psi^*_p$, is called $p$-symmetrical entropy function region. We prove that $\overline{\Psi^*_p}$, the closure of $\Psi^*_p$, is completely characterized by Shannon-type information inequalities if and only if $p$ is the $1$-partition or a $2$-partition with one of its blocks being a singleton. The characterization of the partition-symmetrical entropy functions can be useful for solving some information theory and related problems where symmetry exists in the structure of the problems. Keywords: entropy, entropy function, information inequality, polymatroid., Comment: This paper is published in IEEE Transactions on Information Theory

BATS codes were proposed for communication through networks with packet loss. A BATS code consists of an outer code and an inner code. The outer code is a matrix generation of a fountain code, which works with the inner code that comprises random linear coding at the intermediate network nodes. In this paper, the performance of finite-length BATS codes is analyzed with respect to both belief propagation (BP) decoding and inactivation decoding. Our results enable us to evaluate efficiently the finite-length performance in terms of the number of batches used for decoding ranging from 1 to a given maximum number, and provide new insights on the decoding performance. Specifically, for a fixed number of input symbols and a range of the number of batches used for decoding, we obtain recursive formulae to calculate respectively the stopping time distribution of BP decoding and the inactivation probability in inactivation decoding. We also find that both the failure probability of BP decoding and the expected number of inactivations in inactivation decoding can be expressed in a power-sum form where the number of batches appears only as the exponent. This power-sum expression reveals clearly how the decoding failure probability and the expected number of inactivation decrease with the number of batches. When the number of batches used for decoding follows a Poisson distribution, we further derive recursive formulae with potentially lower computational complexity for both decoding algorithms. For the BP decoder that consumes batches one by one, three formulae are provided to characterize the expected number of consumed batches until all the input symbols are decoded., Comment: 47 pages

In a network, a node is said to incur a delay if its encoding of each transmitted symbol involves only its received symbols obtained before the time slot in which the transmitted symbol is sent (hence the transmitted symbol sent in a time slot cannot depend on the received symbol obtained in the same time slot). A node is said to incur no delay if its received symbol obtained in a time slot is available for encoding its transmitted symbol sent in the same time slot. Under the classical model, every node in a discrete memoryless network (DMN) incurs a unit delay, and the capacity region of the DMN satisfies the well-known cut-set outer bound. In this paper, we propose a generalized model for the DMN where some nodes may incur no delay. Under our generalized model, we obtain a new cut-set outer bound, which is proved to be tight for some two-node DMN and is shown to subsume an existing cut-set bound for the causal relay network. In addition, we establish under the generalized model another cut-set outer bound on the positive-delay region -- the set of achievable rate tuples under the constraint that every node incurs a delay. We use the cut-set bound on the positive-delay region to show that for some two-node DMN under the generalized model, the positive-delay region is strictly smaller than the capacity region., Comment: Submitted to IEEE Transactions on Information Theory in Aug, 2012

Consider a communication network represented by a directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of nodes and $\mathcal{E}$ is the set of point-to-point channels in the network. On the network a secure message $M$ is transmitted, and there may exist wiretappers who want to obtain information about the message. In secure network coding, we aim to find a network code which can protect the message against the wiretapper whose power is constrained. Cai and Yeung \cite{cai2002secure} studied the model in which the wiretapper can access any one but not more than one set of channels, called a wiretap set, out of a collection $\mathcal{A}$ of all possible wiretap sets. In order to protect the message, the message needs to be mixed with a random key $K$. They proved tight fundamental performance bounds when $\mathcal{A}$ consists of all subsets of $\mathcal{E}$ of a fixed size $r$. However, beyond this special case, obtaining such bounds is much more difficult. In this paper, we investigate the problem when $\mathcal{A}$ consists of arbitrary subsets of $\mathcal{E}$ and obtain the following results: 1) an upper bound on $H(M)$; 2) a lower bound on $H(K)$ in terms of $H(M)$. The upper bound on $H(M)$ is explicit, while the lower bound on $H(K)$ can be computed in polynomial time when $|\mathcal{A}|$ is fixed. The tightness of the lower bound for the point-to-point communication system is also proved.

Network coding can significantly improve the transmission rate of communication networks with packet loss compared with routing. However, using network coding usually incurs high computational and storage costs in the network devices and terminals. For example, some network coding schemes require the computational and/or storage capacities of an intermediate network node to increase linearly with the number of packets for transmission, making such schemes difficult to be implemented in a router-like device that has only constant computational and storage capacities. In this paper, we introduce BATched Sparse code (BATS code), which enables a digital fountain approach to resolve the above issue. BATS code is a coding scheme that consists of an outer code and an inner code. The outer code is a matrix generation of a fountain code. It works with the inner code that comprises random linear coding at the intermediate network nodes. BATS codes preserve such desirable properties of fountain codes as ratelessness and low encoding/decoding complexity. The computational and storage capacities of the intermediate network nodes required for applying BATS codes are independent of the number of packets for transmission. Almost capacity-achieving BATS code schemes are devised for unicast networks, two-way relay networks, tree networks, a class of three-layer networks, and the butterfly network. For general networks, under different optimization criteria, guaranteed decoding rates for the receiving nodes can be obtained., Comment: 51 pages, 12 figures, submitted to IEEE Transactions on Information Theory

In a point-to-point communication system which consists of a sender, a receiver and a set of noiseless channels, the sender wishes to transmit a private message to the receiver through the channels which may be eavesdropped by a wiretapper. The set of wiretap sets is arbitrary. The wiretapper can access any one but not more than one wiretap set. From each wiretap set, the wiretapper can obtain some partial information about the private message which is measured by the equivocation of the message given the symbols obtained by the wiretapper. The security strategy is to encode the message with some random key at the sender. Only the message is required to be recovered at the receiver. Under this setting, we define an achievable rate tuple consisting of the size of the message, the size of the key, and the equivocation for each wiretap set. We first prove a tight rate region when both the message and the key are required to be recovered at the receiver. Then we extend the result to the general case when only the message is required to be recovered at the receiver. Moreover, we show that even if stochastic encoding is employed at the sender, the message rate cannot be increased., Comment: Submitted to the IEEE Transactions on Information Theory

Motivated by linear network coding, communication channels perform linear operation over finite fields, namely linear operator channels (LOCs), are studied in this paper. For such a channel, its output vector is a linear transform of its input vector, and the transformation matrix is randomly and independently generated. The transformation matrix is assumed to remain constant for every T input vectors and to be unknown to both the transmitter and the receiver. There are NO constraints on the distribution of the transformation matrix and the field size. Specifically, the optimality of subspace coding over LOCs is investigated. A lower bound on the maximum achievable rate of subspace coding is obtained and it is shown to be tight for some cases. The maximum achievable rate of constant-dimensional subspace coding is characterized and the loss of rate incurred by using constant-dimensional subspace coding is insignificant. The maximum achievable rate of channel training is close to the lower bound on the maximum achievable rate of subspace coding. Two coding approaches based on channel training are proposed and their performances are evaluated. Our first approach makes use of rank-metric codes and its optimality depends on the existence of maximum rank distance codes. Our second approach applies linear coding and it can achieve the maximum achievable rate of channel training. Our code designs require only the knowledge of the expectation of the rank of the transformation matrix. The second scheme can also be realized ratelessly without a priori knowledge of the channel statistics., Comment: 53 pages, 3 figures, submitted to IEEE Transaction on Information Theory

Coherent network error correction is the error-control problem in network coding with the knowledge of the network codes at the source and sink nodes. With respect to a given set of local encoding kernels defining a linear network code, we obtain refined versions of the Hamming bound, the Singleton bound and the Gilbert-Varshamov bound for coherent network error correction. Similar to its classical counterpart, this refined Singleton bound is tight for linear network codes. The tightness of this refined bound is shown by two construction algorithms of linear network codes achieving this bound. These two algorithms illustrate different design methods: one makes use of existing network coding algorithms for error-free transmission and the other makes use of classical error-correcting codes. The implication of the tightness of the refined Singleton bound is that the sink nodes with higher maximum flow values can have higher error correction capabilities., Comment: 32 pages