Shenghao Yang

Coding for a network coded fountain

Batched sparse (BATS) codes are proposed for transmitting a collection of packets through communication networks employing linear network coding. BATS codes generalize fountain codes and preserve the properties such as ratelessness and low encoding/decoding complexity. Moreover, the buffer size and the computation capability of the intermediate network nodes required to apply BATS codes are independent of the number of packets for transmission. It is verified theoretically for certain cases and demonstrated numerically for the general cases that BATS codes achieve rates very close to the capacity of linear operator channels.

Batched Sparse Codes

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 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 and certain multicast networks. For general multicast networks, under different optimization criteria, guaranteed decoding rates for the destination nodes can be obtained.

Weight properties of network codes

In this paper, we first study the error correction and detection capability of codesfor a general transmission system inspired by network error correction. For a given weight measure on the error vectors, we define a corresponding minimum weight decoder. Then we obtain a complete characterisation of the capability of a code for (1) error correction; (2) error detection and (3) joint error correctionand detection. Our results show that if the weight measure on the error vectors is the Hamming weight, the capability of a linear code is fully characterised by a single minimum distance. By contrast, for a nonlinear code, two different minimum distances are needed for characterising the capabilities of the code for error correction and for errordetection. This leads to the surprising discovery that for a nonlinear code, the numberof correctable errors can bemore than half of the number of detectable errors. We also present a framework that captures joint error correction and detection. We further define equivalence classes of weight measures with respect to a channel. Specifically, for any given code, the minimum weight decoders for two different weight measures are equivalent if the two weight measures belong to the same equivalence class. In the special case of linear network coding, we study three weight measures, and show that they are inthe same equivalence class of the Hamming weight and induce the same minimum distance as the Hamming weight. Copyright

Refined Coding Bounds and Code Constructions for Coherent Network Error Correction

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.

Construction of Linear Network Codes that Achieve a Refined Singleton Bound

In this paper, we present a refined version of the Singleton bound for network error correction, and propose an algorithm for constructing network codes that achieve this bound.

Refined Coding Bounds for Network Error Correction

With respect to a given set of local encoding kernels defining a linear network code, refined versions of the Hamming bound, the Singleton bound and the Gilbert-Varshamov bound for network error correction are proved by the weight properties of network codes. This refined Singleton bound is also proved to be tight for linear message sets.

Coding for linear operator channels over finite fields

Linear operator channels (LOCs) are motivated by the communications through networks employing random linear network coding (RLNC). Following the recent information theoretic results about LOCs, we propose two coding schemes for LOCs and evaluate their performance. These schemes can be used in networks employing RLNC without constraints on the network size and the field size. Our first scheme makes use of rank-metric codes and generalizes the rank-metric approach of subspace coding proposed by Silva et al. Our second scheme applies linear coding. The second scheme can achieve higher rate than the first scheme, while the first scheme has simpler decoding algorithm than the second scheme. Our coding schemes only require the knowledge of the expectation of the rank of the transformation matrix. The second scheme can also be realized ratelessly without any priori knowledge of the channel statistics.

Deterministic Secure Error-Correcting (SEC) Network Codes

In this paper, we propose a deterministic algorithm to construct secure error-correcting (SEC) network codes which can transmit information at rate m - 2d - k to all sink nodes, and prevent the information from eavesdropping and contamination during the transmission, where m is the minimum among the maxflows of all the sinks, d is the maximum network Hamming weight of the error vectors and k is the maximum cardinality of the subset of channels which can be eavesdropped. Such constructed network codes can also achieve the refined Singleton bound. Based on this algorithm we further present two transmission schemes which can achieve the transmission rate m - d, when the adversary satisfies an inaction assumption. We also show that in the presence of feedback, a rate beyond m - d could be possibly achieved without reconstructing the existing network code.

Expander graph based overlapped chunked codes

Chunked codes are a variation of random linear network codes with low computational complexities. In chunked codes, the packets in a file are grouped into small (non-overlapped or overlapped) chunks, and random linear encoding operations are performed within each chunk. Previous studies show that when the chunk size is lower bounded by some increasing function of the file length, chunked codes asymptotically achieve the min-cut capacity. However, in most real applications, the chunk size is required to be a small constant due to the computational constraints of network devices. In this case, it remains unknown which rates can be achieved by chunked codes. In this paper, we address the analysis and design of chunked codes with fixed constant chunk sizes. We first highlight the importance of precoding for chunked codes to achieve constant rates, and then present an analysis of non-overlapped chunked (NOC) codes with precoding. We further introduce a new class of chunked codes, called EOC codes, which are based on expander graphs to form overlapped chunks. Numerical and simulation results show that EOC codes achieve significantly higher rates than NOC codes, and also outperform other state-of-the-art overlapped chunked codes.

Finite-length analysis of BATS codes

In this paper, performance of finite-length batched sparse (BATS) codes with belief propagation (BP) decoding is analyzed. For fixed number of input symbols and fixed number of batches, a recursive formula is obtained to calculate the exact probability distribution of the stopping time of the BP decoder. When the number of batches follows a Poisson distribution, a recursive formula with lower computational complexity is derived. Inactivation decoding can be applied to reduce the receiving overhead of the BP decoder, where the number of inactive symbols determines the extra computation cost of inactivation decoding. Two more recursive formulas are derived to calculate the expected number of inactive symbols for fixed number of batches and for Poisson distributed number of batches, respectively. Since LT/Raptor codes are BATS codes with unit batch size, our results also provide new analytical tools for LT/Raptor codes.